Rigidity of Quasiconformal Maps on Carnot Groups

RIGIDITY OF QUASICONFORMAL MAPS ON CARNOT GROUPS

Mark Medwid

A Dissertation

Submitted to the Graduate College of Bowling Green State University in partial fulﬁllment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

August 2017

Committee:

Xiangdong Xie, Advisor

Alexander Tarnovsky, Graduate Faculty Representative

Mihai Staic

Xiangdong Xie, Advisor

Quasiconformal mappings were first utilized by Grotzsch¨ in the 1920’s and then later named by Ahlfors in the 1930’s. The conformal mappings one studies in complex analysis are locally angle-preserving: they map infinitesimal balls to infinitesimal balls. Quasiconformal mappings, on the other hand, map infinitesimal balls to infinitesimal ellipsoids of a uniformly bounded eccentricity. The theory of quasiconformal mappings is well-developed and studied. For example, quasiconformal mappings on Euclidean space are almost-everywhere differentiable. A result due to Pansu in 1989 illustrated that quasiconformal mappings on Carnot groups are almost-everywhere (Pansu) differentiable, as well. It is easy to show that a biLipschitz map is quasiconformal but the converse does not hold, in general. There are many instances, however, where globally defined quasiconformal mappings on Carnot groups are biLipschitz. In this paper we show that, under certain conditions, a quasiconformal mapping defined on an open subset of a Carnot group is locally biLipschitz. This result is motivated by rigidity results in geometry (for example, the theorem by Mostow in 1968). Along the way we develop background material on geometric group theory and show its connection to quasiconformal mappings. iv ACKNOWLEDGMENTS

I would like to acknowledge, first and foremost, my wife, Heather. She’s put up with the long nights and anxiety that accompany graduate school; she constantly cheered me on, told me to be more confident, and agreed to put her life on hold for a number of years while I finished my degree. I’m not sure my finishing graduate school would be possible without her support. Next, I would like to thank my advisor, Xiangdong Xie, who exposed me to a new field of mathematics. He very patiently put up with my na¨ıve questions, ignorance and incorrect proofs; he slowly helped me to become a better mathematician. Beyond academics, he gave me lots of helpful career and life advice that I’ll be sure to apply in the future. He is also one of the first people to encourage me to graduate early – this lead to me accepting a new tenure-track job. I would also like to acknowledge and thank the other members of my dissertation committee (Mihai Staic, Juan Bes´ and Alexander Tarnovsky) for their thoughtful commentary on my manuscript and insightful questions during my defense. Furthermore, I would like to acknowledge my fellow colleagues (both former and present) for their many helpful discussions on life, graduate school and mathematics. I formed many valuable friendships during graduate school. There are too many to be included here, but I will attempt to do so: Jake Laubacher, David Walmsley, Todd Romutis, John Haman and Sam Carolus shared an office with me and kept my day-to-day life more interesting. This was a room filled with exceedingly promising young mathematicians and I was honored to be a part of it. I’d also like to thank Robert Kelvey, Jeff Norton and Leo Pinheiro (who are all now on to greener pastures) for their many helpful conversations with me. My family, in particular, my parents, deserve special mention for all their support over the ten consecutive years I’ve been in school since graduating from high school in 2007. They’ve always had my back, no matter what. Finally, I feel that the Math & Stats Department at BGSU deserves special mention. The department supported my education expenses and my livelihood, and there are so many stellar faculty and staff members that helped form me both as a mathematician and as a teacher. v

TABLE OF CONTENTS Page

CHAPTER 1 INTRODUCTION ...... 1 1.1 The Word Metric ...... 2 1.2 Quasi-Isometries ...... 3 1.3 Quasi-Isometric Rigidity ...... 9

CHAPTER 2 PRELIMINARIES ...... 11 2.1 BiLipschitz Mappings ...... 11 2.2 Quasiconformality ...... 12 2.3 Quasisymmetric Mappings ...... 17

CHAPTER 3 QUASI-ISOMETRIES ON HYPERBOLIC SPACE ...... 24 3.1 The Morse Lemma ...... 24 3.2 The Gromov Boundary ...... 35

CHAPTER 4 CARNOT GROUPS ...... 42 4.1 Lie Algebras and Lie Groups ...... 42 4.2 The Exponential Map & Carnot-Caratheodory´ Distance ...... 46 4.3 Pansu’s Differentiation Theorem ...... 51

CHAPTER 5 MAIN RESULTS ...... 55 5.1 Rigidity of Quasisymmetric Mappings on Fibered Metric Spaces ...... 55 5.2 Rigidity of Quasiconformal Maps on Two-Step Carnot Groups ...... 66

BIBLIOGRAPHY ...... 76 vi

LIST OF FIGURES Figure Page

2.1 An artist’s rendering of what a quasiconformal map might look like...... 13

3.1 The Morse Lemma...... 25 3.2 Two cases for triangles in trees...... 31

3.3 Asymptotic rays in the upper-half plane model of H2...... 36 1

CHAPTER 1 INTRODUCTION

The purpose of this dissertation is to develop the theory accompanying the main result of the author’s research efforts while studying at Bowling Green State University, as well as the proof of the result itself. The author assumes knowledge of basic group theory, analysis, and point-set topology; anything beyond this will be developed in the pages contained hereafter. The following is the author’s main result:

Theorem. Let G = V1 V2 be a two-step Carnot group and let W1 V1 be a subspace and set ✓ W := [W ,W ]. Let W = W W ; W is a Lie sub-algebra of G and so corresponds to a Lie 2 1 1 1 2 subgroup. Suppose that E G is an open set. If f : E G is a quasiconformal mapping that ✓ ! permutes the left cosets of W , then f is locally biLipschitz.

Many of the terms contained in the above theorem are ostensibly foreign to the reader. The basic idea is the following: Carnot groups are spaces that are “nice” enough to carry some notion of differentiability. The definition is fairly similar in flavor to the familiar one from calculus. One then becomes interested in a certain type of (almost everywhere) differentiable function f that we name “quasiconformal.” As in calculus, f needs only to be defined on an open subset for the purposes of the theorem. If f moves a particular subset in a nice enough way, then it turns out that f is locally biLipschitz – to wit, inside a smaller open set, f distorts distances by no more than some bounded factor. This is perhaps enough to remind one of the famed Mean Value Theorem of first year calculus:

Theorem. Let f :[a, b] R be continuous and differentiable on (a, b). Then there exists some ! number c in (a, b) so that

f(b) f(a) = f 0(c) (b a). | | ·

When f has a bounded derivative, then this implies f distorts the lengths of intervals by a bounded factor. The main result of this dissertation is, of course, different in many ways from the 2 Mean Value Theorem, requiring a different setting and assumptions, but the punchline to this quick corollary carries a similar spirit. With the above context, the main theorem is an analysis theorem. However, the study of such theorems also has some deep ties to the world of geometric group theory that will be fleshed out in the body of the dissertation. While quasiconformal analysis is used in several branches of mathematics, it is through geometric group theory that the author was first exposed to the study of quasiconformal mappings. Details for theorems and definitions will be provided (when useful) so as to make this dissertation a stand-alone document for the readers; however, the introductory material is, for the most part, well studied. Many excellent papers and books exist for provid- ing copious details on these subjects – recommendations will be provided when relevant for the interested reader.

1.1 The Word Metric

Recall that a group is finitely generated if there exists a finite set S so that G = S . In what h i follows, unless specified, G denotes a finitely generated group. One can endow G with a metric space structure as follows.

Deﬁnition 1.1.1. (Kapovich, 2013) Suppose that G is a ﬁnitely generated group with generating set S = s1,s2,...,sd . The Cayley graph of G with respect to S is a graph (G, S) with vertex { } set G and edges are constructed according to the rule

there is an edge connecting g and h g = h s✏ for some 1 i d, ✏ 1, 1 . () · i   2{ }

Deﬁnition 1.1.2. (Kapovich, 2013) Let G be a ﬁnitely generated group with generating set S, and suppose (G, S) is the Cayley graph of G with respect to S. Endow with the shortest path metric d – the restriction of d to G, dS, is called the word metric on G with respect to S and is denoted

(G, dS).

One should note that, if e is the identity element of G, then for any g G, d (e, g) is 2 S the reduced word length of g with letters in S. Suppose h, g G; write both h and g as re- 2 3 ✏ ✏ ✏ ✏ duced words with letters in S, i.e. h = s 1h s kh ,g = s 1g s kg . Then the sequence i(1h) ··· i(kh) i(1g) ··· i(kg) ✏1g ✏1g ✏kg 1 (h, hs ,...,hs s ,hg) h hg i(1g) i(1g) i(kg 1) gives an edge-path joining and , so this implies that ··· d(e, g) d(h, hg). Conversely, suppose (h = g ,g ,...,g,g = hg) gives an edge-path joining 0 1 l l+1 1 1 h and hg – then the sequence (e, h g1,...,h gl,g) gives an edge-path joining e and g as gj and g only differ by multiplication on the right by some s", where s S, ✏ 1, 1 . This j+1 2 2{ } illustrates that d(h, hg) d(e, g) and so the two quantities are equal. This means that the word metric on G with respect to S is always left-invariant. With G as a metric space one may talk about the usual notions of limits and continuity of functions. However, one must note that the above definitions depend on the generating set S – different generating sets will produce different Cayley graphs, which in turn will induce different word metrics. One cannot hope for a well-defined “canonical” metric on G, so studying the small- scale metric geometry of G is a futile endeavor. Studying the coarse geometry of G, however, gives a partial solution to the well-definedness issue.

1.2 Quasi-Isometries

Recall that an isometry between metric spaces X and Y is a surjective distance-preserving map f : X Y . If S, T are two different generating sets of G, then there is no reason to suspect that !

(G, dS) and (G, dT ) are isometric. However, they will be “coarsely” isometric in the following sense.

Deﬁnition 1.2.1. (Kapovich, 2013) Let (X, d) and (Y,⇢) be two metric spaces. Let L 1 and A 0. A function f : X Y is an (L, A) quasi-isometry if the following hold: !

1. For any p, q X, 2

1 d(p, q) A ⇢(f(p),f(q)) L d(p, q)+A. L   ·

2. For any y Y , d(f(X),y) A. 2 

Two metric spaces are said to be quasi-isometric if there exists a quasi-isometry between them. 4 A way to summarize the conditions needed for a function to be a quasi-isometry is to say a quasi-isometry is both coarsely biLipschitz and quasi-surjective. A map satisfying the ﬁrst condition is also called an “(L, A) quasi-isometric embedding”. These conditions seem rather innocuous at ﬁrst glance, but in reality a quasi-isometry is poorly behaved – quasi-isometries are not even continuous in general.

Example 1.2.2.

(a) The natural inclusion map Z , R is a (1,1/2) quasi-isometry. !

(b) Let X be any bounded metric space. Then the one-point metric space is quasi-isometric to X. In particular, any ﬁnite group endowed with a word metric is quasi-isometric to the trivial group.

Though quasi-isometries are poorly behaved, they are at least strong enough to carry some structure.

Proposition 1.2.3. The relation given by X Y if and only if there exists a quasi-isometry ⇠ ⇠ from X to Y is an equivalence relation.

Proof. First observe that if X is a metric space, the identity map X X is an isometry. In ! particular, it is a (1, 0) quasi-isometry, so X X. Next, suppose that f : X Y is an (L ,A ) ⇠ ! 1 1 quasi-isometry and g : Y Z is an (L ,A ) quasi-isometry. Then, the composition g f : X Z ! 2 2 ! is also a quasi-isometry. Suppose p, q X. Then, 2

d (g(f(p)),g(f(q))) L d (f(p),f(q)) + A L L d (p, q)+L A + A Z  2 · Y 2  1 2 · X 1 2 1 and the other inequality is similar. Furthermore, if z Z, there exists y Y so that 2 2 d (g(y),z) A , and there exists x X so that d (f(x),y) A . Then, by the triangle Z  2 2 Y  1 5 inequality and the coarse biLipschitz condition,

d (g(f(x)),z) d (g(f(x),g(y)) + d (g(y),z) Z  Z Z L d (f(x),y)+A + A  2 · Y 2 2 L A +2A  2 1 2

Hence, g f is a quasi-isometry and thus X Z. Finally, suppose f : X Y is a (L, A) ⇠ ! quasi-isometry. By quasi-surjectivity, we may construct a “quasi-inverse” to f. Let y Y ; by 2 quasi-surjectivity, there exists x X so that d (f(x ),y) A. Deﬁne g : Y X so that y 2 Y y  ! g(y)=x . Now suppose x X. Since f(x) Y , we have: y 2 2

d (x, g(f(x))) = d (x, x ) L d (f(x),f(x )) + LA 2LA. X X f(x)  · Y f(x) 

This means that g is quasi-surjective. Now, if p, q Y , we have 2

d (g(p),g(q)) = d (x ,x ) L d (f(x ),f(x )) + LA X X p q  · Y p q L(d (f(x ),p)+d (p, q)+d (q, f(x ))) + LA  Y p Y Y q L d (p, q)+3LA.  · Y

For the other inequality, we have

1 A d (g(p),g(q)) = d (x ,x ) d (f(x ),f(x )) . X X p q L Y p q L

Now d (f(x ),f(x )) d (f(x ),p) d (f(x ),p) and d (f(x ),p) d (p, q) d(f(x ),q) Y p q Y q Y p Y q Y q by the triangle inequality, so ﬁnally

1 A 1 3A d (g(p),g(q)) (d (p, q) d (f(x ),p) d (f(x ),q)) d (p, q) . X L Y Y p Y q L L Y L

Hence, g is a quasi-isometry and Y X. ⇠ 6 The above proposition implies that one can consider a metric space up to quasi-isometry. This is not a completely natural notion of equivalence (as, for example, Z, Q, R are all in the same quasi-isometry class). The upside is that considering quasi-isometry classes fixes the dilemma of a well-defined word metric, as it turns out that a (finitely generated) group G equipped with two different word metrics results in metric spaces quasi-isometric to one another. This fact is not entirely trivial. To prove it, we must adopt some basic notions about group actions. We let Sym X denote the set of all bijections X X and Isom X denote the set of all self-isometries of X. ! Definition 1.2.4. Let X be a metric space and G a group and suppose :G Sym X is a group action of G on X. If (G) Isom X, we say that G acts by ! ✓ isometries on X.

1. (Kapovich, 2013) The action of G on X by isometries is said to be properly discontinuous if for any C X compact, the set g G : g(C) C = is ﬁnite. ⇢ { 2 \ 6 ;}

2. (Bridson and Haeﬂiger, 1999) The action of G on X by isometries is said to be cocompact if there exists some compact set C X so that for any x X, there exists some g G so ⇢ 2 x 2 that x g (C). 2 x We say that G acts geometrically on X if the action is both properly discontinuous and cocompact.

Deﬁnition 1.2.5. Let X be a metric space and p, q X. 2 1. If closed balls in X are compact, we say that X is proper.

2. A geodesic joining p and q is an isometric embedding ↵ :[0,d(p, q)] so that ↵(0) = p and ↵(d(p, q)) = q. If any two points may in X may be joined by a geodesic, we say that X is a

geodesic space.

With these deﬁnitions in hand, we may prove the following theorem.

Theorem 1.2.6. (Kapovich, 2013) (The Milnor-Schwarz Lemma) Let G be a group and X be a proper geodesic metric space. If G acts geometrically on X, then G is ﬁnitely generated and quasi-isometric to X when considered as a metric space with a word metric. 7 Proof. Let C be the compact subset of X satisfying the cocompactness condition. Let x C 0 2 and consider B := BR(x0), the closed ball of radius R about x0, with R large enough so that C B (x ). Since X is proper B is compact, and since the action of G is properly discontinuous, ⇢ R 0 the set S = s G : s(B) B = is ﬁnite; write S = s ,s ,...,s . Note here that the identity { 2 \ 6 ;} { 1 2 k} element of G is in S so S must be nonempty. We claim that S generates G. If G = S there is nothing to prove. Consider the quantity

r := inf d(g(B),B):g G S . { 2 \ }

We must have that r>0, since B is compact (in particular, closed) and g(B) is disjoint from B for g S. Now let g G. Connect x to g(x ) by a geodesic and let m be the smallest integer such 62 2 0 0 that d(x ,g(x )) mr + R. Choose points x ,x ,...,x = g(x ) on the geodesic joining x 0 0  1 2 m+1 0 0 and g(x ) so that x B and d(x ,x ) r for 1 j m. Now, for 1 j m, there exists 0 1 2 j j+1      g G so that x g (B), since C B; g is the identity element, as x B. For 1 j m, j 2 j 2 j ⇢ 1 1 2   1 g (x ) B. Furthermore, j j 2

1 1 1 d(g (g (B)),B) d(g (x ),g (x )) = d(x ,x )

1 1 In particular this must mean that B and gj (gj+1(B)) intersect. This implies that gj gj+1 = si(j) for some si(j) in S. This gives:

gm+1 = gmsi(m) = gm 1si(m 1)si(m) = = g1si(1)si(2) si(m) = si(1)si(2) si(m). ··· ··· ···

Hence the set S generates G, and so G is ﬁnitely generated. Now consider G equipped with the word metric relative to S. Then, because of the above calculation, d(g, e) m, where e is the  identity of G. Furthermore by deﬁnition of m,

d(x ,g(x )) R d(x ,g(x )) R 0 0 m 0 0 +1, r   r 8 so we have d(x ,g(x )) R d(g, e) 0 0 +1  r for all g G. Now since the word metric on G is left-invariant, for any h G, 2 2

d(x ,g(x )) R d(h(x ),h(g(x ))) R d(h, hg) 0 0 +1= 0 0 +1,  r r and hence for g ,g G, 1 2 2

d(g (x ),g (x )) R d(g (x ),g (x )) d(g ,g ) 1 0 2 0 +1 1 0 2 0 +1. 1 2  r  r

Now if h = s s s G, then by the triangle inequality, 1 2 ··· k 2

d(x0,h(x0)) d(x0,s1(x0)) + d(s1(x0),s1(s2(x0))) + + d(s1 sk 1(x0))),s1 sk(x0))))  ··· ··· ··· = d(x ,s (x )) + + d(x ,s (x )) , 0 1 0 ··· 0 k 0 2Rd(h, e)  and left invariance again implies for any g ,g G, 1 2 2

d(g (x ),g (x )) 1 0 2 0 d(g ,g ). 2R  1 2

This implies that the map f : G X sending g to g(x ) is coarsely biLipschitz. Since every point ! 0 in X lies in some h(B), we get that every point in X is at most distance R from a point in the image of f, so f is actually quasi-surjective and hence a quasi-isometry.

Corollary 1.2.7. Let G be a ﬁnitely generated group with two different ﬁnite generating sets S and

T . Then (G, dS) and (G, dT ) are quasi-isometric.

Proof. We first consider G as the metric space (G, dS). Since the metric dS is left-invariant, G acts on itself (through left-multiplication) by isometries. Furthermore, the action is geometric. The set e may be moved to any element of G simply by acting on e with the appropriate element, so the { } { } 9 action is cocompact. We next show the action is properly discontinuous. Of course, if G is finite, there is nothing to prove, so assume G is infinite. Suppose the action of G on itself is not properly discontinuous – that is, there exists a compact set C G so that the set g G : g(C) C = ⇢ { 2 \ 6 ;} is infinite. If C is compact, then it is necessarily finite. For any g G, the set g(C) is also finite 2 and g(C) = C . By the finite Pigeonhole Principle there must be one x C that is in g(C) for | | | | 2 infinitely many g G, and again by the finite Pigeonhole Principle, there must be some y C 2 2 so that gy = x for infinitely may g G. This is clearly absurd, and so a contradiction. Hence, 2 the action of G on itself is properly discontinuous – therefore, it is geometric. By Milnor-Schwarz

(and its proof) we may obtain a ﬁnite generating set for G, S0, so that (G, dS) and (G, dS0 ) are quasi-isometric. However the above proof works the same way for the metric space (G, dT ) and we may even obtain the same generating set S0 (using the same R in the proof), so it follows that

(G, dT ) and (G, dS) are quasi-isometric.

The corollary above solves the well-deﬁnedness issue of the group G as a metric space. That is, one may discuss G as a metric space without referencing a generating set as long as we consider equivalence up to quasi-isometry. However, we do lose some information:

Corollary 1.2.8. A ﬁnitely generated group G is quasi-isometric to any of its ﬁnite index subgroups.

Corollary 1.2.9. A ﬁnite group is quasi-isometric to the trivial group.

In particular quasi-isometries do not “see” finite groups or finite subgroups, so we restrict our attention to those groups that are infinite and finitely generated.

1.3 Quasi-Isometric Rigidity

Studying metric spaces up to quasi-isometry means being concerned with properties of the metric space in the large scale. One may wonder in fact what useful properties are preserved up to quasi-isometry. If a property is preserved by quasi-isometry, then it must be “robust” in some sense as (in general) the quasi-isometry is poorly behaved. This is the essence of what is called quasi-isometric rigidity. 10

Definition 1.3.1. (Kapovich, 2013) Let G1,G2 be groups. We say G1 is virtually isomorphic to G if there exist finite-index subgroups H G and finite normal subgroups N /H so that H /N 2 i ⇢ i i i 1 1 is isomorphic to H2/N2.

In light of the Milnor-Schwarz Lemma and its corollaries, one may easily see that two virtually isomorphic groups will be quasi-isometric. The interest is then to reverse this implication.

Deﬁnition 1.3.2. (Kapovich, 2013) Let be a class of groups. We say that the class is quasi- G G isometrically rigid (QI rigid) if whenever a group G is quasi-isometric to some H , there 2G exists G0 so that G is virtually isomorphic to G0. 2G

A way to summarize the above deﬁnition is that the class of groups is QI rigid if whenever a G group G is quasi-isometric to some group in , it follows that G is virtually in . Finding classes G G of groups that are QI rigid is ostensibly a tall order, but it turns out there are many (nice) types of groups that enjoy some sort of QI rigidity. For example, the class of nilpotent groups was shown to be QI rigid by Gromov (1981) as a consequence to his famed theorem on groups of polynomial growth. On the other side, Dyubina (2000) showed that the class of solvable groups is not QI rigid. The proof itself is quite accessible. The study of QI rigidity is just one branch of geometric group theory, but nonetheless, a variety of tools from many areas of mathematics are employed in pursuit of QI rigidity. Furthermore, the concept of the quasi-isometry can be applied in a more analytic setting, as we will see in the subsequent chapters of this dissertation. To appropriately tackle the statement and proof of the author’s main result we need also develop some tools of basic quasiconformal analysis and Pansu differentation. The connection of quasi-isometries to the more famous rigidity theorems, for example the theorems proved by Mostow (1968) or Tukia (1986), will also be explored. 11

CHAPTER 2 PRELIMINARIES

2.1 BiLipschitz Mappings

The main result of this dissertation is concerned with certain types of mappings being locally biLipschitz. In this chapter, we aim to refresh the reader on the basic theory of biLipschitz mappings as well as introducing the concepts of quasiconformality and quasisymmetry.

Deﬁnition 2.1.1. Let (X, dX ) and (Y,dY ) be metric spaces. We say that f : X Y is L-Lipschitz ! (L 1) if, for any x ,x X, we have 1 2 2

d (f(x ),f(x )) L d (x ,x ). Y 1 2  · X 1 2

Roughly speaking, Lipschitz functions distort distances by no more than some uniform factor. We immediately get the following:

Proposition 2.1.2. Let (X, dX ), (Y,dY ) be metric spaces and suppose f : X Y is L-Lipschitz. ! Then f is uniformly continuous.

Proof. Let ">0. Pick = " . Then, if x ,x X are such that d (x ,x ) <, we have L 1 2 2 X 1 2

d (f(x ),f(x )) L d (x ,x )

We should also note that Lipschitz functions in Euclidean space have the following important property, which is often called “Rademacher’s Theorem.”

Theorem 2.1.3. (Heinonen, 2005) Let U Rm and suppose f : U Rn is Lipschitz. Then f is ⇢ ! differentiable almost everywhere.

One may check Heinonen (2005) for a detailed proof of the result, but the main steps in the proof are as follows: show first that the first partial derivatives exist almost everywhere (this gives 12 a candidate for the derivative), then it is shown that the directional derivatives of f in all directions exist almost everywhere, and from there one can show that the derivative is almost everywhere defined.

Deﬁnition 2.1.4. Let (X, dX ), (Y,dY ) be metric spaces. We say that f : X Y is L-biLipschitz ! if, for all x ,x X, we have 1 2 2

1 d (x ,x ) d (f(x ),f(x )) L d (x ,x ). L X 1 2  Y 1 2  · X 1 2

A deﬁnition equivalent to 2.1.4 is that an injective function f : X Y between metric spaces ! 1 is biLipschitz if and only if f and f are Lipschitz. In light of Proposition 2.1.2 one easily sees that a biLipschitz mapping is a homeomorphism (onto its image), and Rademacher’s theorem

1 implies both f and f are almost everywhere differentiable.

2.2 Quasiconformality

Recall from complex analysis that a conformal mapping is an angle-preserving mapping, or one that sends infinitesimal balls to infinitesimal balls. We wish to weaken this condition slightly so that infinitesimal balls are distorted in a somewhat nice way. This concept of “quasiconformal mapping” was first introduced in 1928 by H. Grotzsch¨ which was later given the name “quasiconformal” in the famous work by Ahlfors (1935). Since then, the theory of quasiconformal mappings has been quite well-studied. The excellent set of notes by Vais¨ al¨ a¨ (1971) comprises a good survey of definitions, theorems and detailed proofs, though the more modern notation and exposition of the book by Heinonen (2001) makes a solid introduction to the theory as well – many of the definitions and statements from this chapter will be based on these two texts. Finally, a very short article also by Heinonen (2006) gives a brief but helpful way of building intuition about quasiconformal mappings from various perspectives.

Deﬁnition 2.2.1. (Kapovich, 2013) Let U, V Rn be domains and suppose that f : U V is a ⇢ ! 13

z f f(z)

Figure 2.1: An artist’s rendering of what a quasiconformal map might look like. homeomorphism. Deﬁne the quantity

max x y =r f(x) f(y) Hf (x)=limsup || || || ||. r 0 min x y =r f(x) f(y) ! || || || ||

If there exists some K 0 so that Hf (x) K for all x U, we say that f is K-quasiconformal.  2 If f is K-quasiconformal for some K, we say simply that f is quasiconformal.

One way of summarizing the above definition is that a quasiconformal map sends infinitesimal balls to infinitesimal ellipsoids of uniformly bounded eccentricity. There are a number of other definitions, such as one based on the moduli of curve families, which we will explore subsequently. Before looking at the nice properties of quasiconformal mappings, we will first look at a few simple examples.

Example 2.2.2.

(a) A homeomorphism between domains in Rn is conformal if and only if it is 1-quasiconformal.

(b) The mappings (x, y) (x, y) and (x, y) (x, y) ( 1) are -quasiconformal. 7! 7!

One should note here that (a) in the example above is not entirely trivial. The fact that conformal mappings are 1-quasiconformal is shown in a straightforward manner in the lectures by Vais¨ al¨ a¨ (1971). The reverse direction is more complicated but nevertheless true; 1-quasiconformal mappings in dimension 3 or higher are the restrictions of Mobius¨ transformations. This is due to Gehring (1962). In dimension 2 the result was known from a theorem due to Menshov (1937). 14 The definition given for quasiconformal mappings is rather intuitive in the sense that one can draw a picture of balls getting sent to ellipsoids. However, this definition proves rather unwieldy when it comes to proving anything involving quasiconformal mappings. So, we present an alternative definition:

Deﬁnition 2.2.3. (Vais¨ al¨ a,¨ 1971) Let f : U V be a homeomorphism of domains in Rn, and ! suppose that is a curve family in U (that is, elements of are curves in U). Deﬁne M(), the modulus of the curve family , to be

M() = inf ⇢n dm, ZU where m is the Lebesgue measure on Rn, and the inﬁmum is taken over all nonnegative Borel functions ⇢ : Rn R so that ! ⇢ds 1 Z for all rectiﬁable . 2

Deﬁnition 2.2.4. (Vais¨ al¨ a,¨ 1971) Let f : U V be a homeomorphism of domains in Rn, a ! curve family in U and 0 the image curve family of under f, 0 = f : . Then deﬁne { 2 }

M(0) M() KI (f) = sup ,KO(f)=sup , M() M(0)

where the suprema are taken over all curve families so that M() and M(0) are not simul- taneously 0 or . Set K(f)=maxK (f),K (f) . If K(f) K< , we say that f is 1 { I O }  1 K-quasiconformal; alternatively, f is K-quasiconformal if and only if

M() M(0) K M() K   · for all curve families in U.

It is shown in Vais¨ al¨ a¨ (1971) that the two deﬁnitions are equivalent. The deﬁnition involving 15 the modulus of curve families is (in the author’s opinion) less satisfactory and exceedingly complicated, however, it is more useful in terms of proving statements about quasiconformal mappings. As it turns out, quasiconformal mappings (or QC mappings) have some very nice properties. This is in stark contrast with the other “quasi” mapping discussed, the quasi-isometry. We will see later on that these two types of mappings have a surprising relationship, and this relationship is used in the proofs of some famous rigidity theorems.

Proposition 2.2.5.

(a) Let f : U V and g : V W be homeomorphisms of domains in Rn. If f is K- ! ! quasiconformal and g is K0-quasiconformal, then g f is KK0-quasiconformal.

1 (b) Suppose f : U V be K-quasiconformal. Then f : V U is also K-quasiconformal. ! !

Proof.

(a) Let be a curve family in U, 0 its image family under f, and 00 the image family of 0 under g. Then

1 1 M() M(0) M(00) K0 M(0) KK0 M(), KK0  K0   ·  ·

so g f is KK0-quasiconformal.

1 (b) Let be a curve family in V and suppose that 0 is its image family under f . Then in

particular the image family of 0 under f is , so by quasiconformality of f,

M(0) M() K M(0); K   ·

rearranging the inequality M() K M(0) gives  ·

M() M(0), K  16 M(0) and on the other hand, rearranging the inequality M() gives M(0) K M(), K   · 1 so f is K-quasiconformal.

The theory of quasiconformal mappings is indeed quite well-studied, and we will only need some basic properties as well as one that has a nontrivial proof; see the text by Vais¨ al¨ a¨ (1971) for full details or Heinonen (2005) for a different proof:

Theorem 2.2.6. Let f : U Rn be quasiconformal. Then f is differentiable almost everywhere ! in the sense of Lebesgue measure.

Differentiability is a crucial property of quasiconformal mappings; it is one more tool that makes the theory of quasiconformal mappings particularly nice. Earlier we mentioned a result similar to Theorem 2.2.6 involving biLipschitz mappings. It is here that we should note the relationship between these two types of mappings:

Lemma 2.2.7. Let U, V Rn be domains and suppose f : U V is L-biLipschitz. Then f is ✓ ! L2-quasiconformal.

Proof. Let x U and suppose r>0. If y U is any point so that x y = r, 2 2 || ||

r f(x) f(y) rL. L || || 

1 Hence max x y =r f(x) f(y) rL and min x y =r f(x) f(y) rL , implying || || || ||  || || || ||

max x y =r f(x) f(y) 2 Hf (x) = lim sup || || || || L . r 0 min x y =r f(x) f(y)  ! || || || ||

One should note that if the deﬁnition of quasiconformality were generalized directly to a metric space setting, the same proof for Lemma 2.2.7 works. However, the converse is, in general, false: it is much more difﬁcult to see when a quasiconformal mapping is biLipschitz. 17 Example 2.2.8. Let U = Rn and s> 1. Consider the mapping g(x)= x sx. Then f is || || quasiconformal but is not biLipschitz unless s =0, in which case f is the identity map.

2.3 Quasisymmetric Mappings

Note that the quasiconformal mappings are initially posed as mappings on domains in Eu- clidean space. There is a natural generalization, of course, of the definition to a general metric space. The problem is that quasiconformality is really an infinitesimal condition which may not be interesting in a more general metric space. The idea is then to come up with a “more suitable” definition that more or less agrees with the notion of quasiconformality. This is the notion of a “quasisymmetric” mapping.

Deﬁnition 2.3.1. (Heinonen, 2001) Let f : X Y be a homeomorphism of metric spaces and let ! ⌘ : R+ R+ a homeomorphism. We say that f is an ⌘-quasisymmetry if for all x, y, z X we ! 2 have d (x, y) t d (x, z)= d (f(x),f(y)) ⌘(t) d (f(x),f(z)). X  · X ) Y  · Y

If f is ⌘-quasisymmetric for some ⌘ we say that f is quasisymmetric.

There is also an alternative deﬁnition:

Proposition 2.3.2. Let f : X Y be a homeomorphism of metric spaces X and Y and suppose ! that ⌘ : R+ R+ is a homeomorphism. Then f is ⌘-quasisymmetric if and only if for all ! x, y, z X distinct, we have 2

d (f(x),f(y)) d (x, y) Y ⌘ X . d (f(x),f(z))  d (x, z) Y ✓ X ◆

Proof. Suppose f is ⌘-quasisymmetric and let x, y, z be distinct.

dX (x, y) dX (x, y)= dX (x, z), dX (x, z) · 18 so we get d (x, y) d (f(x),f(y)) ⌘ X d (f(x),f(z)). Y  d (x, z) · Y ✓ X ◆ Now we show the converse. Suppose that x, y, z X and that d (x, y) t d (x, z). Note that 2 X  · X if x = y or x = z there is nothing to prove, so we may suppose without loss of generality that x, y, and z are distinct. Then, we have:

d (f(x),f(y)) d (x, y) Y ⌘ X . d (f(x),f(z))  d (x, z) Y ✓ X ◆

Since ⌘ is a homeomorphism of R+ onto itself, it must be monotone, and hence

d (x, y) t d (x, z) ⌘ X ⌘ · X = ⌘(t), d (x, z)  d (x, z) ✓ X ◆ ✓ X ◆ so d (f(x),f(y)) Y ⌘(t), dY (f(x),f(z))  as desired.

We next will develop a few elementary (but important) properties of quasisymmetric mappings.

1 Proposition 2.3.3. (Heinonen, 2001) Let f : X Y be ⌘-quasisymmetric. Then f is ⌘0- ! quasisymmetric, where 0, if t =0 ⌘0(t)=8 . > 1 < 1 1 , if t>0 ⌘ ( t ) > Proof. Let x, y, z X be distinct; then, since:> f is ⌘-quasisymmetric, 2

d (f(x),f(y)) d (x, y) Y ⌘ X . d (f(x),f(z)  d (x, z) Y ✓ X ◆

This implies

1 dY (f(x),f(y)) dX (x, y) ⌘ . d (f(x),f(z)  d (x, z) ✓ Y ◆ X 19 Therefore, 1 d (x, z) X , ⌘ 1 dY (f(x),f(y)) dX (x, y) dY (f(x),f(z)) ⇣ ⌘ which implies 1 1 dX (f (f(x)),f (f(z))) dY (f(x),f(z)) ⌘0 . d (f 1(f(x)),f 1(f(y)))  d (f(x),f(y)) X ✓ Y ◆ 1 Thus, f is ⌘0-quasisymmetric by the preceding proposition.

Proposition 2.3.4. (Heinonen, 2001) Let X, Y, Z be metric spaces and suppose that f : X Y is ! ⌘ -quasisymmetric and g : Y Z is ⌘ -quasisymmetric. Then g f is (⌘ ⌘ )-quasisymmetric. f ! g g f

Proof. Suppose x, p, q X with 2

d (x, p) t d (x, q). X  · X

Then d (f(x),f(p)) ⌘ (t) d (f(x),f(q)) Y  f · Y as f is a ⌘f -quasisymmetry, and this implies

d (g(f(x)),g(f(p))) ⌘ (⌘ (t)) d (g(f(x)),g(f(q))) Z  g f · Z by the quasisymmetry condition for g.

Proposition 2.3.5. (Heinonen, 2001) The restriction to a subset of an ⌘-quasisymmetric map is again ⌘-quasisymmetric.

Proof. Let X, Y be metric spaces and f : X Y an ⌘-quasisymmetry between them. Let U X ! ⇢ and suppose that x, y, z U with 2

d (x, y) t d (x, z). X  · X 20 Then since f f on U, we have |U ⌘

d (f (x),f (y)) = d (f(x),f(y)) ⌘(t) d (f(x),f(z)) = ⌘(t) d (f (x),f (z)) Y |U |U Y  · Y · Y |U |U by the quasisymmetry condition on f.

The above proofs are simple and use (for the most part) only the deﬁnition, so these proofs may be suitable (depending on one’s taste) as exercises in a ﬁrst course on analysis. Regardless, the above properties are necessary for any practical use of quasisymmetric mappings. With these basic properties in mind we come to an exploration on how a quasisymmetric mapping distorts the relative size of sets.

Proposition 2.3.6. (Heinonen, 2001) Suppose that f : X Y is an ⌘-quasisymmetry and A ! ⇢ B X are such that ⇢ 0 < diam A diam B< .  1

Then, diam f(B) < and we have 1

1 diam f(A) 2diamA ⌘ . 2⌘ diam B  diam f(B)  diam B diam A ✓ ◆ Proof. We will mostly follow the proof given in Heinonen (2001). Pick two sequences (bn), (bn0 ) of elements in B satisfying 1 diam B d (b ,b0 ) 2  X n n and d (b ,b0 ) diam B as n . We will ﬁrst show that f(B) is bounded. For any b B, X n n ! !1 2 we have

d (b, b ) diam B 2 d (b ,b0 ), X 1   · X 1 1 so by the quasisymmetry condition,

d (f(b),f(b )) ⌘(2) d (f(b ),f(b0 )) Y 1  · Y 1 1 21

Similarly, for any b0 B, we have 2

d (f(b0),f(b )) ⌘(2)d (f(b ),f(b0 )); Y 1  Y 1 1 hence, by the triangle inequality,

d (f(b),f(b0)) d (f(b),f(b )) + d (f(b0),f(b )) 2 ⌘(2) d (f(b ),f(b0 )), Y  Y 1 Y 1  · · Y 1 1 which shows that diam f(B) < . Now let x, a A. By the triangle inequality, 1 2

d (b ,b0 ) d (b ,a)+d (b0 ,a). X n n  X n X n

Either d (b0 ,a) d (b ,a) or d (b ,a) d (b0 ,a), so without loss of generality we may X n  X n X n  X n assume d (b0 ,a) d (b ,a) so that d (b ,b0 ) 2 d (b ,a), or X n  X n X n n  · X n

1 d (b ,b0 ) d (b ,a). 2 · X n n  X n

This inequality, together with the quasisymmetry condition, implies

d (f(a),f(x)) d (a, x) diam A 2diamA Y ⌘ X ⌘ ⌘ . d (f(a),f(b ))  d (a, b )  d (a, b )  d (b ,b ) Y n ✓ X n ◆ ✓ X n ◆ ✓ X n n0 ◆

So,

2diamA 2diamA d (f(a),f(x)) ⌘ d (f(a),f(b )) ⌘ diam f(B). Y  d (b ,b ) Y n  d (b ,b ) ✓ X n n0 ◆ ✓ X n n0 ◆

This inequality is true for all n, so since d (b ,b0 ) diam B as n , we have X n n ! !1

2diamA d (f(a),f(x)) ⌘ diam f(B). Y  diam B ✓ ◆

But a and x were arbitrary, so it follows that diam f(A) ⌘ 2diamA . Now, applying this inequality diam f(B)  diam B 22 1 with f and using Proposition 2.3.3 gives

diam A 2diamf(A) ⌘0 , diam B  diam f(B) ✓ ◆ so

diam B 1 diam f(B) ⌘ , diam A 2diamf(A) ✓ ◆ hence diam B diam f(B) ⌘ , diam A 2diamf(A) ✓ ◆ and rearranging this gives the desired second inequality.

Proposition 2.3.6 in particular implies that a quasisymmetry will map bounded sets to bounded sets. Now we see that a map being ⌘-quasisymmetric is a global condition rather than an in- ﬁnitesimal one. What remains is to illustrate that quasisymmetries are in some sense a “correct” generalization of the quasiconformal mappings. Now, based upon the metric deﬁnition of quasiconformality, it is not too hard to see that a quasisymmetric map will be quasiconformal. Indeed, suppose that f : U V is an ⌘-quasisymmetric ! map of domains in Rn and suppose that x U with r>0. Now let y, z U be two points so that 2 2 x y = x z = r. Then by the quasisymmetry condition, || || || ||

f(x) f(y) x y || || ⌘ || || = ⌘(1). f(x) f(z)  x z || || ✓|| ||◆

In particular for any r>0 the quantity

max x w =r f(x) f(w) || || || || min x w =r f(x) f(w) || || || || is bounded uniformly by ⌘(1) so H (x) ⌘(1) for any x U. In particular, f is ⌘(1)-quasiconformal. f  2 The reverse implication is not exactly true: a quasiconformal mapping need not be quasisymmetric. However, quasiconformality, being an inﬁnitesimal condition of sorts, gives local qua- 23 sisymmetry as seen by the following theorem, the “only if” direction of which was given by Vais¨ al¨ a¨ (1981):

Theorem 2.3.7. (Heinonen, 2001) A homeomorphism f : U V between domains in Rn is ! K-quasiconformal if and only if there exists ⌘ so that f is ⌘-quasisymmetric in each open ball

1 B(x, 2 d(x, @U)); here ⌘ depends on K and n. Conversely, K depends on ⌘ and n.

Note that in the instance that U = V = Rn the above theorem reduces to the statement that a mapping f : Rn Rn is quasiconformal if and only if it is quasisymmetric. What one should take ! away from Theorem 2.3.7 is really that a K-quasiconformal mapping is locally quasisymmetric. In Heinonen (2001) it is remarked that the “only if” direction of Theorem 2.3.7 is sometimes referred to as the “Egg Yolk Principle,” although the author has not yet encountered this name “in the wild”, so to speak. As a name, the Egg Yolk Principle is certainly illustrative: supposing that f is quasiconformal in the ball B(x, 2r), then f is ⌘-quasisymmetric in the ball B(x, r). The ball B(x, r) acts as the egg yolk: when the ball B(x, 2r) is hit with f it stretches to some degree (as f is quasiconformal) but the yolk B(x, r) remains relatively intact. Quasisymmetric mappings are a global condition involving the metric of a space X. In particular, it is in some ways easier to work with them than the metric condition for quasiconformality. In fact, quasisymmetric mappings are one of the most useful tools in the proof of the main theorem of this work. 24

CHAPTER 3 QUASI-ISOMETRIES ON HYPERBOLIC SPACE

3.1 The Morse Lemma

In this chapter, we aim to develop some motivation for the study of rigidity of quasiconformal maps. It turns out that quasiconformal mappings are connected to quasi-isometries of certain spaces. The hope is that establishing rigidity properties of quasiconformal maps can give information about rigidity concerning quasi-isometries. One of the main goals of the chapter is to state the famous rigidity theorem by Mostow (1968) and sketch its proof. Another rigidity theorem of Tukia (1986) will also be addressed.

We begin by considering Hn, the real hyperbolic n-space. We identify Hn with the upper half

n n space model (x1,x2,...,xn) R : xn > 0 . In this setting H is a geodesic space, where { 2 } the geodesics are vertical lines or arcs of circles intersecting the horizontal hyperplane along the

n n xn-axis at a right angle. Recall that a geodesic in H is an isometric embedding :[a, b] H . ! There is also an analogue of the geodesic concerning quasi-isometries:

Deﬁnition 3.1.1. (Kapovich, 2013) An (L, A) quasi-isometric embedding :[a, b] X, where ! X is a metric space, is called an (L, A) quasi-geodesic.

We also introduce a notion of distance between sets:

Deﬁnition 3.1.2. (Kapovich, 2013) Let X be a metric space with metric d and suppose that U, V ✓ X. The Hausdorff distance between U and V , dH , is deﬁned as

d (U, V )=inf ">0:U B (V ),V B (U) , H { ✓ " ✓ " } where B (U)= x X : d(x, U) <" , the "-neighborhood about U. " { 2 }

The function dH is not immediately a metric. For example, if U¯ = V¯ then dH (U, V )=0, even if U = V . Furthermore d can be inﬁnite: one only needs to look at two intersecting lines in the 6 H

Euclidean plane to see an example of this. However, dH does at least give some notion of closeness 25

Figure 3.1: The Morse Lemma. between sets. One can make the closed, bounded subsets of X into a metric space equipped with dH , but we will not dwell on this point. Instead, we ﬁrst turn our attention to the fact that, in hyperbolic space, quasi-geodesics are close to geodesics:

Theorem 3.1.3. (Kapovich, 2013) Let L 1,A 0. There exists some number D(L, A) so that n if is an (L, A) quasi-geodesic in H , there exists a geodesic 0 satisfying dH (,0)

The theorem above is sometimes called the Morse Lemma (not to be confused with the Morse Lemma from differential topology), and in Bridson and Haefliger (1999), the theorem is called “Stability of Quasi-Geodesics,” which is perhaps a more descriptive name. To prove Theorem 3.1.3 is somewhat of a long detour, but the approach we adopt follows that of Kapovich (2013). One can see Bridson and Haefliger (1999) for a different method that relies on some delicate geometry about -hyperbolic spaces (the theorem is true in this more general setting; we define this term later in the section). This long diversion is necessary, as Theorem 3.1.3 is one of the key links between quasi-isometries of hyperbolic space to the theory of quasiconformal mappings. We begin first with some notions that come from set theory.

Deﬁnition 3.1.4. (Kapovich, 2013) An ultraﬁlter on N is a surjective function ! :2N 0, 1 !{ } satisfying the following:

!(A B)=!(A)+!(B) !(A B), • [ \

!( )=0. • ; 26 We say that an ultraﬁlter on N is principal if there exists n N so that !( n )=1. Otherwise, if 2 { } ! vanishes on all ﬁnite sets of N, we say ! is nonprinicpal.

A way to compactly state the conditions for ! to be an ultraﬁlter is to simply say that ! is a ﬁnitely additive 0-1 measure on the natural numbers. We can see that ! must be monotone with respect to set inclusion: if B A, then ✓

!(A)=!(B A B)=!(B)+!(A B) !( )=!(B)+!(A B) !(B). [ \ \ ; \

Since ! is surjective there exists some set E with !(E)=1; from this, we obtain that !(N)=1, as well. It also follows that if E N then either !(E)=1or !(Ec)=1. Finally, if !(A)= ✓ !(B)=1, then !(A B) !(A)=!(B) !(A B), [ \ but !(A B)=1, so it follows that !(A B)=1as well. A different characterization of an [ \ ultraﬁlter on N is a collection 2N so that: U⇢ , • ; 62 U

If B and B A, then A , • 2U ✓ 2U

If A, B , then A B , • 2U \ 2U

If A N then either A or Ac . • ✓ 2U 2U

What was shown above is that = A N : !(A)=1 , so the two notions are equivalent. U { ✓ } For the remainder of this discussion on ultrafilters, we will always assume ! to be a nonprincipal ultrafilter. The existence of at least one ! that is nonprincipal is guaranteed by the Axiom of Choice, but we do not know explicitly what ! looks like. The main use of ultrafilters here will be to define convergence of sequences.

Deﬁnition 3.1.5. (Kapovich, 2013) Let X be a Hausdorff topological space and suppose (xn)n1=1 is a sequence in X. We say that the ultralimit of xn is x, written lim! xn = x, if for every 27 neighborhood U of x, we have that !( n N : xn U )=1. That is, almost every xn (in the { 2 2 } sense of !) is in U.

Proposition 3.1.6. (Kapovich, 2013) Let X be a Hausdorff topological space and let (xn)n1=1 be a sequence in X. If limn xn = x, then lim! xn = x also, for any nonprincipal ultraﬁlter !. !1

Proof. Suppose x x as n . Let U be a neighborhood about x. Then there exists some n ! !1 number N so that x U for all n N. Now, !( 1, 2,...,N 1 )=0as ! is nonprincipal, so n 2 { }

!( 1, 2,...,N 1 c)=!( N,N +1,... )=1, { } { }

and since N,N +1,... n N : xn U , !( n N : xn U )=1. { }✓{ 2 2 } { 2 2 }

Lemma 3.1.7. Let (xn), (yn) be sequences of real numbers so that lim! xn and lim! yn exist (as ﬁnite numbers).

(a) If ↵ R, then lim! xn + ↵yn exists and equals lim! xn + ↵ lim! yn. 2

(b) If xn yn for all n N, then lim! xn lim! yn.  2 

Proof. (a) If ↵ =0there is nothing to show. Let x =lim! xn,y =lim! yn, and L = x + ↵y. It

sufﬁces to show that, for any ">0, !( n N : xn + ↵yn (L ",L + ") )=1. Now, let { 2 2 }

" " " " A" := n N : xn x ,x+ ,B" := n N : yn y ,y+ . 2 2 2 2 2 2 2 ↵ 2 ↵ n ⇣ ⌘o ⇢ ✓ | | | |◆

Then by assumption, !(A )=!(B ) = 1. For n A B , " " 2 " \ "

" " x + ↵y

The other inequality follows similarly. Hence,

A" B" n N : xn + ↵yn (L ",L + ") . \ ✓{ 2 2 } 28

It follows that !( n N : xn + ↵yn (L ",L + ") )=1. { 2 2 }

(b) Let x =lim! xn,y =lim! yn. Suppose by way of contradiction that lim! xn > lim! yn.

Choose a R with x (a, ) and y ( ,a). Now !( n N : xn (a, ) )=1, 2 2 1 2 1 { 2 2 1 } and since x y for all n, we have n  n

n N : xn (a, ) n N : yn (a, ) . { 2 2 1 }✓{ 2 2 1 }

c From here, we see that n N : yn ( ,a) n N : yn (a, ) ,a!-measure 0 { 2 2 1 }✓{ 2 2 1 } set, which is a contradiction.

So, the concept of ultralimit is a generalization of the concept of limit in a topological space. We recall from point-set topology the fact that if X is a compact topological space, every sequence has a convergent subsequence. The ultraﬁlter ! is, in some sense, a way of “picking” the convergent subsequence:

Proposition 3.1.8. (Kapovich, 2013) Let X be a compact topological space, (xn)n1=1 a sequence in X and ! a nonprincipal ultraﬁlter. Then (xn) has an ultralimit in X.

Proof. Suppose by way of contradiction that (xn) does not have an ultralimit. That is, around each y X, there exists a neighborhood U of y so that 2 y

!( n N : xn Uy )=0. { 2 2 }

The U cover X, and so we get a ﬁnite subcover U ,U ,...,U . Then the sets y { y1 y2 yk }

n N : xn Uyj { 2 2 } all have !-measure 0, a contradiction, as they union to make N, which has !-measure 1.

Corollary 3.1.9. If (xn) is a bounded sequence in R, then lim! xn exists (and is finite). 29 Of course, the ultralimit in the proposition above depends on the ultrafilter !. However, we will simply consider a fixed nonprincipal ultrafilter ! for the remainder of these discussions. The concept of “ultralimit” manifests in the proof of the Morse Lemma through ultralimits of sequences of metric spaces.

Deﬁnition 3.1.10. (Kapovich, 2013) Let (Xn,dn) be a sequence of metric spaces. For (xn), (yn) 2

(Xn), deﬁne

d!((xn), (yn)) = lim dn(xn,yn). !

Identify the sequences (x ) via the equivalence relation (x ) (y ) if and only if d ((x ), (y )) = n n ⇠ n ! n n 0. Choose a sequence of base points (pn) (Xn). Deﬁne lim! Xn = X!, the ultralimit of the 2 sequence of spaces Xn, to be the equivalence classes (xn) of sequences in (Xn) so that

d ((x ), (p )) < . ! n n 1

One can check that the distance d! is well-deﬁned and that X! indeed forms a metric space. Our principal interest in ultralimits of metric spaces is using the same space X, but rescaled for all n.

Deﬁnition 3.1.11. (Kapovich, 2013) Let (n) be a sequence of positive numbers so that lim! n =

0, (X, d) a metric space, and (pn) a sequence of points in X. The asymptotic cone of X,

Cone (X), is deﬁned to be lim!(X,nd, pn), where nd is the metric d rescaled by n.

Naturally, Cone (X) depends on !, (pn) and (n), but for sake of notation, we will assume Cone (X) to mean that these quantities have been chosen in the deﬁnition. One particularly nice use of the asymptotic cone is having quasi-isometries induce much nicer mappings:

Proposition 3.1.12. (Kapovich, 2013) Let Xn,Yn be sequences of metric spaces, (pn) a sequence of points where p X , ( ) a sequence of positive scalars so that lim =0, and f : n 2 n n ! n n X Y a sequence of (L, A) quasi-isometric embeddings. Deﬁne X⇤,Y⇤ to be the spaces n ! n n n 30

Xn,Yn respectively with metric rescaled by n. Then (fn) induces an L-biLipschitz mapping f : X Y , where X =lim X⇤ and Y =lim Y ⇤. ! ! ! ! ! ! n ! ! n

Proof. Suppose that (an), (bn) X!. For n N we have: 2 2

n d (a ,b ) A d (f(a ),f(b )) Ld (a ,b )+ A. L Xn n n n  n Yn n n  n Xn n n n

Taking ultralimits and applying Lemma 3.1.7 yields

1 d ((a ), (b )) d ((f(a )), (f(b ))) Ld ((a ), (b )). L X! n n  Y! n n  X! n n

This shows that the mapping

f : X Y ,f((x )) = (f(x )) ! ! ! ! ! n n

is an L-biLipschitz embedding of X! into Y!.

The case that Xn = X and Yn = Y and fn = f for some ﬁxed (L, A) quasi-isometric embedding reduces to f inducing a biLipschitz mapping of Cone (X) into Cone (Y ). Moreover, it is not difﬁcult to see that if the fi from the proposition above were quasi-isometries (that is, also quasi-surjective), then f! will be onto – in particular, a homeomorphism. So, quasi-isometries induce rather well-behaved maps on asymptotic cones. This fact is made all the more surprising when we remember that quasi-isometries aren’t necessarily continuous! We get the following as a corollary, which follows after one shows that Cone (Rn) is isometric to Rn:

Corollary 3.1.13. Rn and Rm are quasi-isometric if and only if m = n.

To prove the Morse Lemma, we need some preparatory deﬁnitions and lemmas.

Deﬁnition 3.1.14. (Bridson and Haeﬂiger, 1999) Let X be a geodesic space and let x, y, z X. 2 Denote by [x, y] the image of the geodesic joining x and y. Then the geodesic triangle T =

[x, y] [x, z] [y, z] is said to be -slim, >0, if each edge of T is contained in the union of [ [ 31 x y z

x y

Figure 3.2: Two cases for triangles in trees. neighborhoods of the other two edges of T . If every geodesic triangle in X is -slim, we say that

X is -hyperbolic. When the is not important we say that X is simply (Gromov) hyperbolic. Finally, if X is -hyperbolic for any >0, X is said to be 0-hyperbolic.

Lemma 3.1.15. A geodesic space X is 0-hyperbolic if and only if it is a tree.

Proof. Suppose X is a tree. In a tree, without loss of generality, there are two possibilities for geodesic triangles: either the three vertices make three distinct geodesic segments, or they only make one (see Figure 3.2). In either case, each geodesic segment in the triangle is contained in a union of the other two segments. In particular, these triangles are 0-slim. Hence X is 0- hyperbolic. Conversely, suppose that X is 0-hyperbolic. Recall a topological arc in X is a subset homeomorphic to the unit interval. By Mayer and Oversteegen (1990), it sufﬁces to show that X is uniquely arcwise connected (that is, any two points in X are connected by a unique arc) with convex distance function (the distance d being convex in this setting means that there exists a topological arc joining x and y, A, so that for any z A, d(x, y)=d(x, z)+d(z,y)) Note 2 that X is already geodesic so the distance function in X is automatically convex. We show that any two points x and y in X can be joined by a unique geodesic. Suppose by contradiction that there are two points x and y with two or more geodesics joining them; denote these by [x, y]1 and [x, y] . There exists z [x, y] and z0 [x, y] so that z = z0. Let [x, z], [z,y] [x, y] , 2 2 1 2 2 6 ⇢ 1 [x, z0], [z0,y] [x, y] and suppose [z,z0] is a geodesic joining z and z0. I claim one of the geodesic ⇢ 2 triangles [x, z] [x, z0] [z,z0] or [z,y] [z0,y] [z,z0] is not 0-slim. Otherwise, we would have [ [ [ [ [z,z0] [x, z] [x, z0] and [z,z0] [z,y] [z0,y]. Since [z,z0] [x, z] [x, z0] and z = z0, each of ⇢ [ ⇢ [ ⇢ [ 6 [x, z] and [x, z0] must contain subgeodesics of [z,z0]. So, suppose [p, z] [x, z] [z,z0]. Similarly, ⇢ \ 32 suppose [z,q] [z,y]; this means that the geodesic [x, y] is backtracking, a contradiction, as ⇢ 1 distance-preserving curves are always injective.

A basic fact about Hn is that the triangles have uniformly bounded area. In particular this means Hn is -hyperbolic for some appropriate choice of , so the triangles in Hn are “slim” in the sense deﬁned above, but there is another notion of “thin triangles” that is also satisﬁed by Hn:

Deﬁnition 3.1.16. (Bridson and Haeﬂiger, 1999) Let X be a geodesic space and suppose that T

2 is a geodesic triangle in X with vertices x, y, z. Let T 0 be a triangle in R with vertices x0,y0,z0 so that d(x, y)= x0 y0 ,d(y, z)= y0 z0 , and d(x, z)= x0 z0 (T 0 is called a comparison || || || || || || triangle for T in R2). Then T is said to satisfy the CAT(0) inequality if, for any two p, q T and 2 their comparison points p0,q0 T 0, we have that d(p, q) p0 q0 . X is said to be a CAT(0) 2 || || space if every geodesic triangle satisﬁes the CAT(0) inequality.

In other words, a CAT(0) space is a space in which triangles are thinner than their appropriate comparison in Euclidean space. A key property of CAT(0) spaces is that they are uniquely geodesic: this is easy to see because if two geodesics join the points x, y X, then pick a point 2 on either geodesic, and this forms a geodesic triangle, T . The comparison triangle in this case is actually degenerate since R2 is uniquely geodesic and hence the two geodesics must coincide. One can show that Hn is CAT(0); this is actually not too difficult to see given the more general definition of a CAT() space for  R, found in Bridson and Haefliger (1999), as Hn is actually 2 CAT( 1), which implies CAT(0). We shall not delve much into the theory of CAT(0) spaces except for the following fact:

Lemma 3.1.17. (Kapovich, 2013) Let (Xn,dn,pn) be a sequence of pointed CAT(0) spaces. Then

X! =lim! Xn is again CAT(0).

Proof. A geodesic in X is given by a sequence of geodesics ↵ where ↵ X . Hence a triangle ! n n ⇢ n 2 T! is given by a sequence of triangles Tn; consider the comparison triangle T!0 in R along with 33 the comparison triangles T 0 for the triangles T . Two points (a ), (b ) T satisfy n n n n 2 !

d (a ,b ) a0 b0 n n n || n n|| because the triangles T satisfy the CAT(0) inequality, so taking ultralimits yields that d ((a ), (b )) n ! n n  a0 b0 , where a0,b0 are the appropriate comparison points for (a ) and (b ). || || n n

In particular, if X is CAT(0), Cone (X) is CAT(0) as well. We use this fact to establish the following:

Lemma 3.1.18. (Kapovich, 2013) Every asymptotic cone Cone (Hn),n 2, is a tree.

Proof. Note that since Cone (Hn) is CAT(0) it is uniquely geodesic and hence any triangle T in

n n Cone (H ) is determined as the ultralimit of a sequence of triangles Tk in (H ,kd). There is a

n n uniform for which Tk, viewed as a triangle in H , is -slim, so as a triangle in (H ,kd), Tk

n is k-slim. But lim! k =0, so the triangle T is 0-slim; this means that Cone (H ) is a tree by Lemma 3.1.15.

There is but one more lemma to prove before we move to the proof of the Morse Lemma:

Lemma 3.1.19. (Kapovich, 2013) Let X be a tree and suppose that ↵ :[0, 1] X is continuous ! and injective. Then after a reparametrization ↵ is a geodesic arc in X.

Proof. Let ↵ be continuous and injective and suppose that ↵0 is the geodesic connecting ↵(0) and

↵(1). Suppose n N. We consider the piecewise geodesic 2

n (n) (n) ↵n := [xk 1,xk ], k[=1

(n) (n) (n) where x Im ↵ and x = ↵(0),xn = ↵(1). This serves as an “approximation” of the arc k 2 0 ↵, as d (Im ↵ , Im ↵) 0 as n , where d denotes Hausdorff distance, for appropriately H n ! !1 H (n) chosen x . I claim that for all n, Im ↵0 Im ↵ . This is due to 0-slimness: for n =2, the triangle k ✓ n (2) (2) (2) (2) (2) (2) [x ,x ] [x ,x ] [↵(0),↵(1)] is 0-slim, meaning that ↵0 =[↵(0),↵(1)] [x ,x ] 0 1 [ 1 2 [ ✓ 0 1 [ 34 (2) (2) [x1 ,x2 ]. Inductively, suppose that the image of ↵0 is contained in the image of ↵n 1. Then, by 0-slimness, [x(n),x(n)] [x(n),x(n)] [x(n),x(n)], 0 2 ✓ 0 1 [ 1 2 so we simply replace [x(n),x(n)] [x(n),x(n) with [x(n),x(n)], but then this becomes a piecewise 0 1 [ 1 2 0 2 geodesic with n 1 pieces, so applying the induction hypothesis gives that Im ↵0 Im ↵ for all ✓ n n 2. In particular, this means that Im ↵0 is contained in Im ↵. Now by the Intermediate Value 1 Theorem the map ↵ ↵0 is onto, so it must be the case that Im ↵ and Im ↵0 are actually equal.

At long last, we return to the proof of the Morse Lemma.

Proof of Theorem 3.1.3. We ﬁrst prove the Morse Lemma in the case that the quasi-geodesics are

n n of ﬁnite length. Suppose that ↵ :[0,a] H is an (L, A) quasi-geodesic and let ↵0 :[0,a0] H ! ! be the geodesic joining ↵(0) and ↵(a). Deﬁne the quantities

D↵ := dist(↵,↵0):=sup d(↵(t), Im ↵0),D↵0 := dist(↵0,↵):= sup d(↵0(t), Im ↵). t [0,↵] t [0,a ] 2 2 0

Now d (Im ↵, Im ↵0)=max D ,D ; we shall show that D and D are uniformly bounded H { ↵ ↵0 } ↵ ↵0 over all (L, A) quasi-geodesics ↵, but since the proof for D↵0 is completely analogous, we only show it for D↵. We proceed by contradiction: we get a sequence of (L, A) quasi-geodesics ↵i :

n n Ji := [0,ai] H (and, also, a sequence of geodesics ↵i0 : Ji0 := [0,ai0 ] H deﬁned in the same ! ! manner as before for D ) so that D as n . For each i, choose x ↵ (J ) so that ↵ ↵i !1 !1 i 2 i i

1 d(x , Im ↵0) D < . | i i ↵i | i

1 n Furthermore, set i = . Now, rescale the metrics on Ji,Ji0 and H by i; taking ultralimits D↵i n gives, by Proposition 3.1.12, an L-biLipschitz embedding ↵! : J! Cone (H ) and an isometric ! n embedding ↵!0 : J!0 Cone (H ). The sets J! and J!0 are isometric to either closed intervals or ! the set [0, ); furthermore, since ↵ and ↵0 have comparable lengths, it follows that J is of ﬁnite 1 i i ! length if and only if J!0 is. Moreover, dist(↵!,↵!0 )=1, by choice of the points xi and the fact that 35 we rescale at each i by i. We have two cases to consider.

Suppose ﬁrst that J! is of ﬁnite length. Since ↵i and ↵i0 shared endpoints for all i, so too do ↵!

n and ↵!0 . By Lemma 3.1.18, Cone (H) is a tree, and so by Lemma 3.1.19, the images of ↵! and

↵!0 are the same. But this is a contradiction, as dist(↵!,↵!0 )=1. Next, suppose that J is inﬁnite, that is, J =[0, ). Remember that the quasi-ray (without ! ! 1 the additive constant A) ↵! and ray ↵!0 are still within unit distance from one another. Let x! be

n the point in Cone (H ) represented by (xi); let C = d(↵!(0),x!) (note that x! is in the image of ↵! since xi was in the image of ↵i for all i). Since dist(↵!,↵!0 )=1, we may ﬁnd a geodesic

connecting points x = ↵!(s),x0 = ↵!0 (s0) of length at most 1 so that d(↵!(0),x) >C, and

Im Im ↵ = x , Im Im ↵0 = x0 , for some s, s0 [0, ). Since d(↵ (0),x) >C, the \ ! { } \ ! { } 2 1 ! arc given by = ↵ ([0,s]) Im includes the point x . But shares endpoints with the geodesic ! [ ! 0 = ↵0 ([0,s0]), contradicting Lemma 3.1.19, as x 0. ! ! 2 \

Hence, the quantity D↵ must be uniformly bounded, and by an analogous argument, D↵0 is as well. Hence, the result is true for quasi-geodesics of finite length. For quasi-geodesics of infinite length, in other words, ↵ :[0, ) Hn or ↵ : R Hn, we simply look at the restriction of ↵ to 1 ! ! finite subintervals of arbitrary length and apply the finite-length result.

The Morse Lemma is also true when one replaces Hn with a Gromov hyperbolic space, as mentioned earlier. The above approach will work. There is an alternate proof presented in Bridson and Haefliger (1999) completely different in flavor; this proof just requires some knowledge about -hyperbolic spaces, but the approach we adopted above serves more to give the flavor of other interesting concepts in geometric group theory – namely ultrafilters and ultralimits, -hyperbolic spaces, and CAT(0) spaces.

3.2 The Gromov Boundary

In this section we describe the Gromov boundary of a -hyperbolic space along with its relationship to quasi-isometries of the underlying space. Informally speaking, this is the set of all points “at inﬁnity” that a ﬁxed observer can see. 36

Figure 3.3: Asymptotic rays in the upper-half plane model of H2.

Deﬁnition 3.2.1. (Bridson and Haeﬂiger, 1999) Suppose that X is a -hyperbolic space and ,0 :

[0, ) X are geodesic rays. Then we say that the rays ,0 are asymptotic if dH (Im ,Im 0) 1 ! is ﬁnite.

Equivalently, and 0 will be asymptotic if and only if the quantity supt [0, ) d((t),0(t)) is 2 1 finite. It is not too difficult to see that the relationship of rays being asymptotic is an equivalence relation. With this fact in mind, we get the following definition.

Definition 3.2.2. (Bridson and Haefliger, 1999) Let X be Gromov hyperbolic. Then define @X, the Gromov boundary of X, to be the equivalence classes of asymptotic geodesic rays. For a geodesic ray , we write its equivalence class as ( ). 1

However, in a proper Gromov hyperbolic space X, we may regard the Gromov boundary as rays emanating from a single ﬁxed base point:

Proposition 3.2.3. (Bridson and Haeﬂiger, 1999) Let X be a proper geodesic -hyperbolic space and p X. For any ⇠ @X, there exists a geodesic ray :[0, ) X so that (0) = p and 2 2 1 ! ( )=⇠. 1

Proof. Let ↵ :[0, ) X be a geodesic ray with ↵( )=⇠. We shall show that there is a 1 ! 1 geodesic ray starting at p asymptotic to ↵. So, for n N, we consider n to be a geodesic joining 2 p to ↵(n). Let R = d(↵(0),p); in the closed (compact, since X is proper) ball B¯(p, Re2) we get (1) ¯ 2 a subsequence of n, nk , converging uniformly on compact subsets of B(p, Re ) by the Arzela-´ (1) (2) Ascoli Theorem. Then we may obtain a further subsequence of nk , nk , converging uniformly 37 on compact subsets of B¯(p, Re3). Then, repeating inductively, we ﬁnd a further subsequence (i 1) (i) ¯ i+1 of nk , called nk , converging uniformly on compact subsets of B(p, Re ). Then we get (i) that the sequence (ni )i1=1 will converge uniformly on compact subsets of X to a geodesic ray :[0, ) X. To see that ( )=↵( ), we note that the piecewise geodesic [↵(0),p] Im 1 ! 1 1 [ n is a quasi-geodesic with the same endpoints as the geodesic segment ↵([0,n]); hence, by the Morse Lemma (for -hyperbolic spaces) there is some D>0 so that the Hausdorff distance between

Im n and Im ↵([0,n]) is less than D, for all n; it follows that and ↵ must be asymptotic.

Note that hyperbolicity condition is really only used in the proposition above to invoke the Morse Lemma. Nevertheless, we imagine the Gromov boundary to be, in some sense, the directions in which a ﬁxed observer can look (assuming that “looking” in this context is really a geodesic ray emanating from a ﬁxed point in X). One can describe the topology on @X as follows:

Definition 3.2.4. (Bridson and Haefliger, 1999) Let X be a proper -hyperbolic space. A generalized ray is a geodesic :[0,R] X, where R [0, ]. In the case that R< we also define ! 2 1 1 (t)=(R) for t [R, ]. Hence we may view X @X as the set of generalized rays starting 2 1 [ at a fixed point p X. Then, define convergence in X¯ = X @X to be x x as n if and 2 [ n ! !1 only if there is a sequence of generalized rays so that (0) = p and ( )=x converging n n n 1 n uniformly on compact subsets to a generalized ray with (0) = p and ( )=x. 1

So long as X is proper and -hyperbolic, this topology does not depend on the choice of base point p. Furthermore the natural inclusion X, X¯ is actually a homeomorphism, so this deﬁnition ! in some sense gives back the “original” topology on X. We consider @X as a topological space endowed with the subspace topology.

The deﬁnition of the Gromov boundary may seem hard to visualize, but in the case of Hn+1, n 1, it is rather easy to visualize. In the ball model of Hn+1, where geodesics are circular arcs orthogonal to the boundary of the ball, one can see that geodesics will be asymptotic if and only if they approach the same limit point on the boundary sphere. So, there is a natural mapping

@Hn+1 Sn that is actually a homeomorphism. Therefore, we will consider @Hn+1 and Sn to be ! 38 synonymous.

Now, suppose that f : Hn+1 Hn+1 is an (L, A) quasi-isometry, and let be a geodesic ! ray. Then f is an (L, A) quasi-geodesic ray. By the Morse Lemma, there is a geodesic ray 0 asymptotic to the quasi-geodesic ray f . If ⇠ = ( ) and ⇠0 = 0( ), then we see that the 1 1 n+1 (L, A) quasi-isometry f has induced a mapping on the boundary of H by sending ⇠ to ⇠0.

Deﬁnition 3.2.5. (Kapovich, 2013) Let n 1 and f : Hn+1 Hn+1 be an (L, A) quasi-isometry. ! n n+1 For any ⇠ S , let be a geodesic ray in H so that ( )=⇠. Let 0 be a geodesic ray that 2 1 is D-Hausdorff close to f ; this number D and ray 0 are provided by the Morse Lemma. Then deﬁne f , the boundary extension of f, to be the mapping 1

n n f : S S ,f(⇠)=0( ). 1 ! 1 1

It seems a priori that f should have a well-deﬁnedness issue. However, f (⇠) does not 1 1 n 1 depend on the choice of the geodesic ray ; it only depends on the point ⇠ S . Suppose ↵, 2 are two rays so that ⇠ = ↵( )=( ). Then d (Im ↵, Im )=M< . Then, since f is an 1 1 H 1 (L, A) quasi-isometry, d (Im f ↵, Im f ) LM + A and so d (Im ↵0, Im 0) < , which H  H 1 gives that ↵0( )=0( ). 1 1 Quasi-isometries are, of course, poorly behaved, but we saw in the discussion on asymptotic cones that they could be “nicely” behaved maps in ultralimit. There is no reason to expect f to 1 even be continuous, as it is induced by a quasi-isometry. We shall see, however, that the opposite is in fact true: such boundary extensions are actually very nice maps, as well.

Theorem 3.2.6. (Kapovich, 2013) Let n 1 and f : Hn+1 Hn+1 be an (L, A) quasi-isometry. ! Then the boundary extension f : Sn Sn is an ⌘-quasisymmetry, where ⌘ depends only on n, L, 1 ! and A.

To prove the above, we ﬁrst establish a lemma, which says that quasi-isometries “quasi-commute” with nearest-point projections: 39 Lemma 3.2.7. (Kapovich, 2013) Let f : Hn+1 Hn+1 be an (L, A) quasi-isometry and suppose ! n+1 is a geodesic ray in H . Let 0 be a geodesic ray that is D-Hausdorff close to f . Deﬁne n+1 n+1 ⇡ : H Im and ⇡0 : H Im 0 to be the nearest-point projections onto and 0, ! ! respectively. Then there exists C = C(L, A) so that, for all x Hn+1, 2

d(f(⇡(x)),⇡0(f(x))) C. 

Proof of Theorem 3.2.6. We prove ﬁrst that f is a homeomorphism. Suppose is a geodesic ray 1 n+1 in H . Let ⇠ = ( ). Choose any sequence (xi) in the image of so that limi xi = ⇠. Then, 1 !1 1 letting ⇡ denote the nearest-point projection onto the image of , we consider the spaces ⇡ (xi).

1 n The boundary of these subspaces ⇡ (xi) forms a sphere in S , and this sphere bounds a round

n ball B S – this B contains the point ⇠. Let 0 denote the geodesic ray close to f(), as in the i ⇢ i lemma. Let y = f(x ) for i 1, 2,... . Then the quasi-isometry condition of f says i i 2{ }

1 d((0),x) A d(f((0)),y) L i  i for all i, and since d((0),x) as i , the sequence (y ) cannot be bounded. But the quasi- i !1 !1 i geodesic f() is Hausdorff-close to 0, so limi yi =limt 0(t)=⇠0 = f (⇠). By the lemma, !1 !1 1 1 1 we see that f(⇡ (xi)) is contained in a C = C(L, A)-neighborhood of ⇡0 (⇡0(yi)) and hence

n f (Bi) is contained in a ball Bi0 S , where ⇠0 Bi0. Now the Bi0 form a basis for the topology 1 ⇢ 2 n of S at ⇠0, so this in fact gives that f is continuous by the local formulation of continuity. Also, 1 f is onto by quasi-surjectivity of f. To see injectivity suppose ⇠,⇣ Sn with f (⇠)=f (⇣). 1 2 1 1 n+1 Let ⇠,⇣ be rays in H with ⇠( )=⇠ and ⇣ ( )=⇣. Then dH (⇠,⇣ ) < ; consider a 1 1 1 1 1 1 1 quasi-inverse f for f. Since f is a quasi-isometry f (⇠) and f (⇣ ) are within bounded Hausdorff distance, and these quasi-rays are within bounded distance of the rays limiting to ⇠ and ⇣, hence ⇠ = ⇣. Since Sn is compact Hausdorff, this gives that f is a homeomorphism. 1 We now consider the upper-half space model of Hn+1. It sufﬁces to show that f is quasisym- 1 metric in the case that is the vertical geodesic ray above 0=⇠, f (⇠)=0, and 0, the ray 1 40 supplied by the Morse lemma, is . Further, we also assume f ( )= . Now, we consider the 1 1 1 topological annulus

n A = x R : R1 d(0,x) R2 , { 2   } where 0

d := log R log R =log(R /R ). 2 1 2 1

Now by the lemma, d(⇡ f,f ⇡ ) C,  so f(A) is contained in an interval of length c := 2C + Ld + A. In particular this means that  f (A) is contained in the annulus 1

n A0 = x R : R10 d(0,x) R20 , { 2   } where R2 ec. Deﬁne ⌘ :[1, ) [e2C+A, ) as R1  1 ! 1

⌘(t)=e2C+L log t+A = tLe2C+A.

We see that ⌘ is continuous, monotone and surjective. Now any distinct x, y, z Rn will determine 2 d(x,y) an annulus A, and the eccentricity of A0 f(A) is bounded by ⌘( ), so this gives that f is d(x,z) 1 ⌘-quasisymmetric.

Corollary 3.2.8. Let f : Hn+1 Hn+1 be an (L, A) quasi-isometry. Then f : Sn Sn is ! 1 ! quasiconformal.

This establishes the connection of the earlier mentioned theory of quasi-isometries and quasi- isometric rigidity with the theory of quasiconformal mappings. This relationship has been used in geometry when proving rigidity results.

Deﬁnition 3.2.9. Let G be a group of quasiconformal mappings on Sn. If there is a constant 41 K 1 so that every g G is K-quasiconformal, we call G a uniform quasiconformal group. 2

One important result by Tukia (1986) is the following:

Theorem 3.2.10. Let G be a countable uniform quasiconformal group acting on Sn (n 2), so that almost every point of Sn is a conical limit point. Then G can be conjugated by a quasiconformal map f into a conformal group.

We have not addressed the term “conical limit point.” One may consult Kapovich (2013) for the full details, but it essentially boils down to the following idea: suppose G “acts” via quasi- isometries on a -hyperbolic space X. Suppose that for ⇠ @X, there exists a ray limiting to 2 ⇠, a point x X, and a sequence (g ) (with g G) so that g x converges to ⇠ in an R-cone 2 i i 2 i · about ; we then call ⇠ a conical limit point. A key element in the proof of the above is the ability to translate between quasi-isometries of Hn+1 and quasiconformal mappings of Sn (its Gromov boundary). This idea was also utilized earlier on in the famous theorem of Mostow (1968).

Deﬁnition 3.2.11. Let G be a group of isometries acting freely and properly discontinuously on

Hn. Then we call Hn/G a hyperbolic manifold.

Theorem 3.2.12. Suppose that M and N are ﬁnite-volume hyperbolic manifolds of dimension n 3, and f : ⇡ (M) ⇡ (N) is an isomorphism. Then f is induced by a unique isometry from 1 ! 1 M to N.

Rigidity results in geometry and group theory are those of the following ﬂavor: Property A implies Property B, but Property B does not imply Property A in general; with Property B and some mild assumptions, Property B implies (something close to or exactly) Property A. For example, in Mostow’s Theorem above, Property A is that of being isometric, while Property B is that of having isomorphic fundamental group. When it comes to quasi-isometric rigidity, Property A is that of being virtually isomorphic, while Property B is that of being quasi-isometric. 42

CHAPTER 4 CARNOT GROUPS

4.1 Lie Algebras and Lie Groups

The last bit of background to consider is the theory of Carnot groups, a particular type of Lie group. To begin, we recall some elementary facts and deﬁnitions about Lie groups.

Deﬁnition 4.1.1. Let V be a real vector space, and suppose [ , ]:V V V is a map. We say · · ⇥ ! that (V,[ , ]) is a Lie algebra if the following hold: · · (a) [ , ] is bilinear: For all x, y, z V and ↵, R, we have: · · 2 2

[↵x + y,z]=↵[x, z]+[y, z], [x, ↵y + z]=↵[x, y]+[x, z].

(b) V is anti-commutative: For all x, y V , 2

[x, y]= [y, x].

[[x, y],z]+[[y, z],x]+[[z,x],y]=0.

If V is a Lie algebra, we call the associated map [ , ] the Lie bracket of V . · ·

Example 4.1.2. (a) Consider gln(R), the set of n n matrices with real entries. Equip gln(R) ⇥ with the matrix commutator bracket:

[A, B]=AB BA,

where the product AB is understood to be matrix multiplication. Then, with the matrix

commutator bracket, gln(R) is a Lie algebra. 43 (b) Let V = R3 and consider the following operation:

[(x ,x ,x ), (y ,y ,y )] = (x y x y ,x y x y ,x y x y ), 1 2 3 1 2 3 2 3 3 2 3 1 1 3 1 2 2 1

that is, [x, y]=x y, where here is the cross product of two vectors that one learns about ⇥ ⇥ in calculus. Then R3 equipped with the cross product as a bracket is a Lie algebra.

Lie algebras themselves are interesting objects of study. In a graduate course in algebra, for example, one may learn about the classiﬁcation of semisimple Lie algebras via Dynkin diagrams and root systems. This topic has a nice geometric ﬂavor, but has no overt relevance to this particular paper. For those interested, consult the standard reference text by Carter (1972). Instead, we will examine the relationship between Lie algebras and their associated Lie groups.

Deﬁnition 4.1.3. (Warner, 1983) A Lie group G is a smooth manifold and a group so that the map

1 (g, h) gh is smooth (C1). 7!

So, Lie groups are groups with some sort of calculus structure. Remember that a vector ﬁeld

X on Rn is an assignment of vectors v Rn to vectors of the form 2

n @ X(v)= f (v) , i @x i=1 i X

n where xi are the standard coordinates and fi : R R. If each component fi is smooth, then X ! is said to be a smooth vector field. One can define a vector field on a manifold analogously.

Definition 4.1.4. (Warner, 1983) Let (G, ) be a Lie group and suppose h G. Then left trans- ⇤ 2 lation by h is the mapping Lh : G G defined by Lh(g)=h g. A vector field X on G is called ! ⇤ left invariant if dLhX = XLh.

The correspondence between Lie algebras and Lie groups is given by the following:

Proposition 4.1.5. (Warner, 1983) Let G be a Lie group and denote by g the set of all left-invariant vector ﬁelds on G. Then g is isomorphic as a vector space to the tangent space at the identity TeG. 44 Proof. First we note that g is a real vector space. Deﬁne F : g T G by F (X)=X(e). If ! e X, Y g and ↵ R then 2 2

F (X + ↵Y )=(X + ↵Y )(e)=X(e)+↵Y (e)=F (X)+↵F (Y ), so F is linear. Moreover, F is injective: indeed, if F (X)=F (Y ) then X(e)=Y (e) and for any h H, 2

X(h)=X(Lh(e)) = dLh(X(e)) = dLh(Y (e)) = Y (Lh(e)) = Y (h), so X = Y . We next show F is surjective. Let v T G. For g G, deﬁne X(g)=dL (v). Then 2 e 2 g

F (X)=X(e)=dLe(v)=v.

Furthermore, X is left invariant. If g, h G, then 2

dLh(X(g)) = dLh(dLg(v)) = dLhg(v)=X(hg)=X(Lh(g)), by the Chain Rule. Therefore F is a linear isomorphism.

Now, one can equip g with the bracket [X, Y ]=XY YX, giving the following result, the proof of which we leave as an exercise, as the details are tedious and difﬁcult to typeset:

Proposition 4.1.6. (Warner, 1983) Let G be a Lie group and g its vector space of left invariant vector ﬁelds. Then (g, [ , ]) is a Lie algebra, where [ , ] is the bracket described above. · · · ·

Deﬁnition 4.1.7. Let G be a Lie group and g the Lie algebra of left invariant vector ﬁelds of G. We say that g is the Lie algebra associated to G.

We showed above that g is isomorphic (as a vector space) to TeG. We also deﬁne a notion of isomorphism between Lie algebras:

Deﬁnition 4.1.8. Let g, h be Lie algebras. A function f : g h is said to be a morphism of Lie ! 45 algebras if f is linear and respects the bracket operations of g and h: that is, for x, y g, 2

f([x, y]g)=[f(x),f(y)]h.

Moreover, if f is an isomorphism of vector spaces and a morphism of Lie algebras, we say that f is a Lie algebra isomorphism.

The isomorphism from g to TeG induces a Lie algebra structure on TeG, so it is sometimes convenient to view the Lie algebra associated to G as the tangent space of G at the identity.

Example 4.1.9. Let H be the group of 3 3 matrices of the form ⇥

1 xz 0 1 01y . B C B C B001C B C @ A There is an obvious bijection : R3 H where !

1 xz 0 1 (x, y, z)= 01y . B C B C B001C B C @ A We equip R3 with the group law (one can verify that this does, indeed, form a group):

(x, y, z) (x0,y0,z0)=(x + x0,y+ y0,z0 + xy0 + z). ⇤

It is clear from matrix multiplication that :(R3, ) (H, ) is a homomorphism, and hence an ⇤ ! · isomorphism. Furthermore,

1 (x, y, z) =( x, y, z + xy), 46 1 and so the map sending ((x, y, z), (x0,y0,z0)) (x, y, z) (x0,y0,z0) =(x x0,y y0,z + 7! ⇤ x0y0 xy0 z0) is smooth. In particular, this means H is a Lie group. It is called the Heisenberg group. One can show that its Lie algebra h is isomorphic to the 3-dimensional real vector space with basis x, y, z and bracket relations [x, y]=z, [x, z]=[y, z]=0. { }

If one believes that h from Example 4.1.9 is the Lie algebra with basis and bracket relations as above then there is a sort of “grading” of subspaces, that is, if V1 denotes the subspace of h generated by x and y and V2 denotes the subspace of h generated by z, then we have

h = V V , 1 2 with [V ,V ] V . As it turns out, there is a class of Lie algebras with this property. 1 1 ⇢ 2

Deﬁnition 4.1.10. Let g be a Lie algebra. If there exist nontrivial subspaces V1,V2,...,Vk so that

g = V V V 1 2 ··· k

and [V1,Vi]=Vi+1, where 1 i k and Vk+1 := 0 , then we say that g is a k-step Carnot   { } Lie algebra. If G is a simply connected Lie group so that its Lie algebra g is a k-step Carnot Lie algebra, then we say that G is a k-step Carnot group.

4.2 The Exponential Map & Carnot-Caratheodory´ Distance

To a Lie group one may associate a Lie algebra (and, it turns out, the converse is also true), but one may be curious as to how this fact can be applied. In the example of the Heisenberg group H we noted the group operation and the bracketing relations of the associated Lie algebra h. We will presently explore the relationship between the group law of H and the bracket structure of h.

Definition 4.2.1. (Warner, 1983) Let M be a manifold and X a smooth vector field on M.We say that a smooth curve : U M (U R open) is an integral curve of X if, for every ! ✓ t U,0(t)=X((t)). 2 47 Now it turns out that in the case that the manifold M in the definition above is a Lie group and

X is a left invariant vector field, then there is a unique integral curve : R M for X so that ! (0) = e (one may consult Warner (1983) for complete details). This induces a mapping between left-invariant vector fields and elements of the Lie group, which is made precise with the following definition:

Deﬁnition 4.2.2. (Warner, 1983) Let G be a Lie group and g its Lie algebra. For any X g let 2 X : R G denote the unique integral curve for X so that X (0) = e. Deﬁne !

exp : g G !

as exp X = X (1). The mapping exp is called the exponential map on g.

In short, the exponential map gives us a way to translate from a Lie algebra to an operation. However, we are usually more interested in the reverse direction – that is, giving some sort of representation of the group multiplication law in terms of Lie brackets. There is some hope of being able to do this, as evidence by the following theorem, the proof of which we will not provide here.

Theorem 4.2.3. (Warner, 1983) Let G be a Lie group with Lie algebra g. Then exp : g G is ! smooth and, when restricted to a neighborhood of 0 g, is a diffeomorphism onto a neighborhood 2 of e in G.

This theorem shows that, at least in a local sense, it is possible to recover the group operation of G in terms of the bracket operation of g. Now, let U g and V G be neighborhoods of 0 ⇢ ⇢ and e, respectively, so that f := exp : U V is a diffeomorphism. Let X, Y U be such |U ! 2 1 that f(X) f(Y ) V . Then, define X Y = f (f(X) f(Y )), which becomes a new “group ⇤ 2 ⇤ ⇤ multiplication law.” The idea of pulling back the product of two points X, Y U via the inverse of exponentiation 2 sounds like a perfectly nice theoretical idea. It turns out that this idea is more practical than 48 it seems, because there is an actual formula for computing the new group operation. This is the famous Baker-Campbell-Hausdorff (BCH) formula. As one might expect, it is a rather intimidating creature, but its significance is not really the formula itself. The existence of this formula is what is significant. The following combinatorial formula is due to Dynkin (1947):

n 1 r1 s1 rn sn 1 ( 1) [X Y X Y ] X Y = ··· (4.2.4) ⇤ n n (r + s ) n r !s ! n=1 r +s >0,1 i n i=1 i i i=1 i i X i i X  · P Q which has been simpliﬁed with the following notation:

[Xr1 Y s1 Xrn Y sn ]=[X, [X, [X, [Y,[Y, [Y , [X, X, [X, [Y,[Y, Y ], ], ··· ··· ··· ··· ··· ··· ··· r1 times s1 times rn times sn times | {z } | {z } | {z } | {z } and the sum runs over all possible integer combinations ri + si > 0. Note that anti-commutativity of the bracket implies [X, X]=[Y,Y ]=0for any X, Y in the Lie algebra, so the expression above reduces to 0 if sn > 1 or if sn =0and rn > 1.

Deﬁnition 4.2.5. Let g be a Lie algebra. For X g, deﬁne ad X : g g as ad X(Y )=[X, Y ]. 2 ! We say that g is nilpotent if there exists some n N so that, for every X1,X2,...,Xn g, 2 2

ad X ad X ad X 0. 1 2 ··· n ⌘

In particular, all Carnot Lie algebras are nilpotent. Since the bracket eventually vanishes, for nilpotent Lie algebras, the expression in 4.2.4 is actually ﬁnite. For example, in the case that [X, Y ]=0, the BCH formula becomes X Y = X + Y . For the Lie algebra associated to the ⇤ Heisenberg group H, for example, we get X Y = X + Y + 1 [X, Y ]. So for nilpotent Lie ⇤ 2 algebras (for example, Carnot Lie algebras) the BCH formula is ﬁnite everywhere and not just in a neighborhood of 0. In fact we can say even more:

Theorem 4.2.6. (Corwin and Greenleaf, 1990) Let G be a simply connected nilpotent Lie group with Lie algebra g. Then the exponential map exp : g G is a diffeomorphism. ! 49 This means that in the case of nilpotent Lie groups the bracket structure of the Lie algebra completely determines the group operation, which can be extremely useful for computation.

Example 4.2.7. For the Heisenberg group H, we recall that h has a basis X, Y, Z obeying the { } bracket relations [X, Y ]=Z, [X, Z]=[Y,Z]=0. Hence the group law “ ” on h becomes: ⇤

(aX + bY + cZ) (dX + eY + fZ)=(a + d)X +(b + e)Y +(c + e)Z ⇤ 1 + [aX + bY + cZ, dX + eY + fZ] 2 1 =(a + d)X +(b + e)Y +(c + e + (ae bd))Z. 2

We can now freely use the group law given by the BCH formula when working with Carnot groups. However, in doing analysis on Carnot groups, it is necessary to establish a metric on them.

We would like this metric dC on Carnot groups G to have the following properties:

1. dC induces the usual topology on G.

2. d is left-invariant: for x, g, h G, d (g, h)=d (x g, x h). C 2 C C ⇤ ⇤

The metric dC is constructed as follows. Identify the k-step Carnot group G with its Lie algebra g via the exponential map. So, g = V V V , 1 2 ··· k where the V are nontrivial subspaces of g and [V ,V] V . Now V consists of (by definition) i 1 i ⇢ i+1 1 left-invariant vector fields on G. Then for p G, define 2

H G = X(p):X V . p { 2 1}

We have H G is a subspace of T G for all p G. This is the horizontal subspace at p. p p 2 Deﬁnition 4.2.8. Let G be a Carnot group. A piecewise smooth curve c :[a, b] G is said to be a ! horizontal curve if c0(t) Hc(t)G for almost every t [a, b] (in the sense of Lebesgue measure). 2 2 We also have the following, independently due to Chow (1939) and Rashevskii (1938): 50 Theorem 4.2.9. Let G be a connected Carnot group; then any two points in G may be joined by a horizontal curve.

Deﬁnition 4.2.10. Let G be a Carnot group with Lie algebra g = V1 Vk. Fix a norm ··· |·| on V . Then gives a norm on the horizontal subspaces of G; for g, h G deﬁne the length of a 1 |·| 2 horizontal curve c :[a, b] G to be !

b l(c)= c0(t) dt. | | Za

Finally, for g, h G, deﬁne 2

d (g, h)=inf l(c):c is a horizontal curve from g to h . C { }

We say that dC is the Carnot-Caratheodory´ metric on G.

It is easy to see that d (g, h)=d (h, g) for all g, h G. Moreover, if x, g, h G, and c ,c C C 2 2 1 2 are horizontal curves joining g to x and x to h, respectively, then the concatenation of c1 and c2 is a horizontal curve joining g and h and so it follows that d (g, h) d (g, x)+d (x, h). Furthermore C  C C if d (g, h)=0then g = h. The Chow-Rashevskii Theorem guarantees that d (g, h) < for C C 1 all g, h G, so d is a metric on G. Furthermore, d satisfies the two desired conditions on the 2 C C previous page. A metric on G is a wonderful thing to have, but the downside is that there is no formula for the distance between two elements of G. We instead look at an alternative “distance” on g, which is diffeomorphic to G by the exponential map. Supposing g = V V , we fix norms on 1 ··· k |·|(i) V . Then, for X = X + X + + X , where X V , define i 1 2 ··· k i 2 i

k X = X 1/i = X + X + + k X . || || | i|(i) | 1|(1) | 2|(2) ··· | k|(k) i=1 X q q 51 Then, via the BCH formula, set

d(X, Y )= ( X) Y . || ⇤ ||

It turns out that d is biLipschitz to dC , so practically speaking, we will use “d” for computations rather than dC .

4.3 Pansu’s Differentiation Theorem

The last item to address in regards to the theory of Carnot groups is the notion of so-called “Pansu derivatives.” Let G be a Carnot group with associated algebra g = V V . For ease 1 ··· k of notation, write X = X + X = k X 1/i, where X V and X refers to the || || || 1 ··· k|| i=1 | i| i 2 i | i|

ﬁxed norm on Vi. Suppose we want a linear mapP that dilates the norm of X by t>0:

: g g, (X) = t X . t ! || t || || ||

Then,

k k k t X = t X 1/i = t X 1/i = tiX 1/i = tX + t2X + + tkX . || || | i| | i| | i| || 1 2 ··· k|| i=1 i=1 i=1 X X X So deﬁning k k i t(X)=t Xi = t Xi i=1 ! i=1 X X gives us the desired dilation property. Now if g is, for example, a 2-step Carnot Lie algebra, then

[X1 + X2,Y1 + Y2]=[X1,Y1 + Y2]+[X2,Y1 + Y2]

=[X1,Y1]+[X1,Y2]+[X2,Y1]+[X2,Y2]

=[X1,Y1], 52 2 so t([X1 + X2,Y1 + Y2]) = t [X1,Y1]=[tX1,tY1]. However,

2 2 [t(X1 + X2),t(Y1 + Y2)] = [tX1 + t X2,tY1 + t Y2]

2 2 2 2 =[tX1,tY1]+[tX1,t Y2]+[t X2,tY1]+[t X2,t Y2]

=[tX1,tY1],

so this gives that t([X, Y ]) = [t(X),t(Y )]. Naturally, one can also do this computation for k- step Carnot Lie algebras. This means that the dilation t is a Lie algebra morphism; in fact, it is an isomorphism. Since the BCH formula is deﬁned in terms of brackets it follows that, if X, Y g, 2

(X Y )= (X) (Y ). t ⇤ t ⇤ t

In particular, d(t(X),t(Y )) = td(X, Y ). Now since G is a Carnot group, the exponential map is a diffeomorphism, so if g G, there 2 exists a unique X g so that exp X = g. Hence, if we wanted to deﬁne a similar on G, it would 2 t make sense to have

t(g)=exp(t(X)).

and so we use the same name t because the diagram

t G G

exp exp

g g t

commutes. It follows from the BCH formula that (g h)= (g) (h) for all g, h G. t ⇤ t ⇤ t 2 53

This also makes t a similarity with respect to the Carnot-Caratheodory´ distance, namely

dC (t(g),t(h)) = tdC (g, h).

Now, of particular interest are mappings that commute with t:

Definition 4.3.1. (Le Donne and Xie, 2016) Let g, h be two Carnot Lie algebras and suppose that f : g h is a Lie algebra morphism. We say that f is a strata-preserving homomorphism if ! f = f . t t We are now ready to define the notion of Pansu differentiability. It is similar in spirit to the definition of a map f : Rn Rm being differentiable, which we recall here: !

Deﬁnition 4.3.2. Let f : Rn Rm be a function. We say that f is differentiable at v if there ! exists a linear map L : Rn Rm so that !

f(v + h) f(v) Lh m lim || ||R =0, h 0 h n ! || ||R and this map L is called the derivative of f at v.

This deﬁnition says that in some sense differentiability at a point means that the function is approximately linear at that point. The Pansu derivative is, naturally, attributed to Pansu (1989):

Deﬁnition 4.3.3. Let G, H be Carnot groups, U G, V H open sets and f : U V a ⇢ ⇢ ! function. We say that f is (Pansu) differentiable at g U if there exists a strata-preserving 2 homomorphism L : G H so that !

1 1 d ((f(g)) f(h),L(g h)) lim C ⇤ ⇤ =0. h g ! dC (g, h)

In this case the function L is called the Pansu derivative (or Pansu differential) of f at g.

Similar to the deﬁnition of differentiability in the real case, the above deﬁnition says that a function is Pansu differentiable at a point if it is approximately a strata-preserving homomorphism. 54 Pansu (1989) proved the following, a Carnot group analogue of Theorem 2.1.3 (Rademacher’s Theorem):

Theorem 4.3.4. (Pansu’s Differentiability Theorem) Let G, H be Carnot groups and U G, V ⇢ ⇢ H open sets. If f : U V is quasiconformal, then f is almost-everywhere Pansu differentiable. ! Furthermore, the Pansu derivative is almost-everywhere a strata-preserving isomorphism.

In particular, biLipschitz mappings are almost-everywhere Pansu differentiable. We note that there is perhaps some confusion about the measure with respect to which Theorem 4.3.4 means by “almost everywhere.” The Lebesgue measure on the underlying Carnot Lie algebra g is, as it turns out, a left-invariant measure on g, and so under identiﬁcation with the exponential map we can obtain a left-invariant measure on G; it is a left-invariant Haar measure. 55

CHAPTER 5 MAIN RESULTS

5.1 Rigidity of Quasisymmetric Mappings on Fibered Metric Spaces

We begin the chapter on results with a rigidity-type result on fibered metric spaces; the case in which the quasi-symmetry is globally defined is discussed in Le Donne and Xie (2016). We first introduce some new terms:

Deﬁnition 5.1.1. (Le Donne and Xie, 2016) Let X be a metric space; we say two closed subsets U and V of X are parallel if there exists some constant c>0 so that d(u, V )=d(v, U)=c for all u U, v V . 2 2

Definition 5.1.2. (Le Donne and Xie, 2016) Let (X, d) be a metric space and L 1. We say that X is an L-fibered metric space if X admits a covering by closed sets (and members of are U U called fibers) with the following properties:

1. Each ﬁber U is L-biLipschitz to an unbounded geodesic space.

2. Parallel ﬁbers are non-isolated: for any U , there exists a sequence of ﬁbers U so 2U i 2U

that U and Ui are distinct, parallel and limi dH (U, Ui)=0. !1

3. Distinct ﬁbers have positive distance: for any U, V distinct, d(U, V ) > 0, where d(U, V ) 2U is the usual distance between closed sets.

4. Non-parallel ﬁbers diverge: if U, V are not parallel, then d (U, V )= . 2U H 1

If X is a metric space satisfying conditions 1, 3 and 4, we shall say X is an L-ﬁbered metric space without non-isolated parallel ﬁbers.

Deﬁnition 5.1.3. Let X, Y be metric spaces. Given K 1 and C>0 we say that a bijection f : X Y is a (K, C)-quasi-similarity if !

C d (x ,x ) d (f(x ),f(x )) CK d (x ,x ) K X 1 2  Y 1 2  · X 1 2 56 for all x ,x X. 1 2 2 It is clear that a mapping is a quasi-similarity if and only if it is biLipschitz. However the point is that the notion of quasi-similarity offers more information. In particular one can often have some degree of control over K but not C. In Le Donne and Xie (2016), the following is proved:

Theorem 5.1.4. Let X, Y be L-fibered metric spaces and suppose f : X Y is an ⌘-quasisymmetry ! sending fibers of X homeomorphically onto fibers of Y . Then f is a (K, C)-quasi-similarity, where K depends only on ⌘ and L.

The ﬁrst result we would like to illustrate is one that is similar to the above. However, ﬁrst we introduce some more preliminary terms:

Deﬁnition 5.1.5. Let f : X Y be a map between metric spaces. Then deﬁne the upper ! pointwise Lipschitz constant of f at x X to be 2

dY (f(x),f(y)) Lf (x)=limsup . y x dX (x, y) !

Similarly, the lower pointwise Lipschitz constant of f at x X is 2

dY (f(x),f(y)) lf (x)=liminf . y x ! dX (x, y)

In Heinonen (2001), Lf (x) is simply referred to as the pointwise Lipschitz constant of f at x.

However, since we will have occasion to use both quantities Lf (x) and lf (x), it was necessary to distinguish the two quantities.

Deﬁnition 5.1.6. Let X, Y be metric spaces and f : X Y an ⌘-quasisymmetry. Let x X. If ! 2 0

L (x) Kl (x), f  f we call x a K-good point with respect to f. For U X, if we may ﬁnd a constant K = K(⌘) 1 ⇢ 57 so that x is K-good with respect to f for all x U, we call the points of U uniformly f-good 2 with f-good constant K.

The ﬁrst result is as follows:

Theorem 5.1.7. Let X, Y be L-fibered metric spaces (possibly without non-isolated parallel fibers). Suppose in addition there exists a continuous function C :(0, ) [1, ) so that if U is a fiber 1 ! 1 of X and x, y U then 2

1 d(y, U˜) d(x, U˜) C(d(x, y)) d(y, U˜) (?) C(d(x, y))   · for all other fibers U˜, and Y also satisfies condition (?) with C˜. Let E X be open. Suppose ⇢ f : E Y is an ⌘-quasisymmetry that satisfies the following: !

1. f maps fibers homeomorphically to fibers, or more precisely, for each fiber U of X, f maps each component of U E homeomorphically onto an open subset of a fiber of Y . \

2. Every ﬁber has a dense set of uniformly f-good points, with f-good constant K.

Then f is a local (M,C)-quasi-similarity, where M is quantitative in ⌘,C and the f-good constant K.

The case where all fibers of X are parallel satisfies condition (?) with C =1. Furthermore, we note there are settings in which assumption 2 is achieved. For instance, in the case of Carnot groups, quasisymmetric mappings are Pansu differentiable almost-everywhere. In particular this means almost-all points in a Carnot group are uniformly f-good with constant K = ⌘(1). Assuming this theorem, we first give an example illustrating a situation in which it can be used. This example is used in Le Donne and Xie (2016) to show that the conclusion of their theorem fails if one replaces the condition “parallel fiber” with the weaker condition that the Hausdorff distance between them is finite. The theorem above, however, applies to this example as we do not assume that parallel fibers are non-isolated. 58 Example 5.1.8. Consider X = Y = C with the usual metric. For fixed ↵> 1 and ↵ =0, the 6 map z z ↵z is quasiconformal. We consider the mapping 7! | |

z ↵z z 1 f(z)=8| | | | . > <>z z 1 | | > :> In X, the fibers are simply horizontal lines (in particular here C =1). In Y the fibers are the images of fibers of X under f (outside the unit ball these are again horizontal lines). Note that Y also satisfies condition (?) if f is restricted to a neighborhood away from the origin. The function f is quasiconformal (hence quasisymmetric) hence differentiable almost-everywhere, giving a dense set of uniformly f-good points. So, f is locally biLipschitz (by the theorem) if f is restricted to an open set away from the origin.

The following lemma will be used in proofs of both of the author’s theorems:

Lemma 5.1.9. Let X be a geodesic space and Y a metric space. Suppose f : X Y is a map so ! that L (x) b f  for all x X. Then f is Lipschitz. 2

To prove it, we recall the Lebesgue Number Lemma from topology:

Lemma 5.1.10. (Munkres, 2000) Let A be an open covering of the metric space (X, d). If X is compact, there exists >0 such that for each subset of X having diameter less than , there exists an element of A containing it.

Proof. Let A be an open covering of X. If X is in A there is nothing to prove, so assume X A . Let A ,...,A be a ﬁnite subcover of X; this is guaranteed by compactness of X. Set 62 { 1 n}

Ci = X Ai and define \ n 1 f(x)= d(x, C ). n i i=1 X 59 Note that the map x d(x, C ) is continuous for all i, so f is continuous as well. Since X is 7! i compact the Extreme Value Theorem says that f attains a minimal value. We first show this value is necessarily positive. Let x X. Since A ,...,A covers X, x A for some 1 i n. 2 { 1 n} 2 i   Openness of A means we can find ">0 so that B(x, ") A . This means that d(x, C ) ", i ⇢ i i forcing f(x) > 0. Let =minf(x). x X 2 Now, suppose that B X has diameter less than . Let x B. Let 1 M n so that ⇢ 0 2   d(x0,CM )=max1 i n d(x0,Ci). Since B has diameter less than , B B(x0,). Furthermore,   ⇢ we have f(x ) d(x ,C ).  0  0 M

Since d(x ,C ) , B(x ,) A , and therefore, B A . 0 M 0 ⇢ M ⇢ M

We now give the proof for Lemma 5.1.9.

Proof of Lemma 5.1.9. Let p, q X with p = q. Consider the geodesic ↵ :[0,d(p, q)] X 2 6 ! joining p and q. For any x ↵, there exists a neighborhood U of x so that 2 x

d(f(z),f(x)) 2b d(z,x) 

for all z Ux. The collection Ux x ↵ forms an open cover of ↵ and hence we get a ﬁnite subcover 2 { } 2 U ,U ,...,U . Now by the Lebesgue Number Lemma, we have >0 so that for ⇠,⇣ ↵ { x1 x2 xn } 2 with d(⇠,⇣) <, then ⇠,⇣ U for some i. So, pick y [0,d(p, q)] with 2 xi i 2

0=y

and yi yi 1 <for 1 i k. Then d(↵(yi 1),↵(yi)) <for 1 i k, so by the triangle | |     60 inequality

d(f(p),f(q)) d(f(p),f(↵(y1))) + + d(f(↵(yk 1)),f(q))  ···

2b d(p, ↵(y1)) + +2b d(↵(yk 1),q)  · ··· · =2b d(p, q). · Hence, f is 2b-Lipschitz.

The proof of Theorem 5.1.7 is presented in a series of lemmas: ﬁrst we show the result is true when f is restricted to a ﬁber. We then use this fact to establish the result in an open set containing the point we are interested in.

Lemma 5.1.11. Let U be any ﬁber in X and suppose that B is a bounded open subset of U. Furthermore, suppose x U is a K-good point with respect to f. For any y B, there exists an 0 2 2 open ball B˜ B so that f ˜ is an (M,l (x ))-quasi-similarity; M is quantitative in ⌘, K and the ⇢ |B f 0 diameters of B and f(B).

Proof. Let y B and x be a K-good point with respect to f on U. We begin with an observation: 2 0 for z U sufficiently close to y, we may find another fiber U˜ with the property that d(y, z)= 2 z d(y, U˜z). Indeed, suppose w is a point in X so that d(y, w) >d(y, z) and w is in the same path- component of X as y. Join y and w with a continuous path ↵ :[a, b] X. Note that for any ! t [a, b] there exists a fiber U with ↵(t) U . Then let 2 ↵(t) 2 ↵(t)

t =inf t [a, b]:d(y, U ) >d(y, z) , 0 { 2 ↵(t) } and let U˜ = U , so that d(y, z)=d(y, U˜ ). Let y˜ U˜ be the point at which d(y, U˜ ) is z ↵(t0) z z 2 z z achieved, that is, d(y, U˜ )=d(y, y˜ ). Similarly, let x˜ U˜ be the point at which d(f(x ),f(U˜ )) z z z 2 z 0 z is achieved and let x U be such that d(x ,x )=d(x , x˜ ). Now as z y, x x . Hence we z 2 0 z 0 z ! z ! 0 may choose z close enough to y so that xz satisﬁes

d(f(xz),f(x0)) 2Lf (x0). d(xz,x0)  61 So for any such z, quasi-symmetry of f gives

d(f(y),f(z)) d(f(y),f(z)) d(f(y),f(˜y )) ⌘(1) z . d(y, z)  d(y, y˜z)  · d(y, y˜z)

Now d(f(y),f(˜y )) is comparable to d(f(y),f(U˜ )): choose y0 U˜ so that z z 2 z d(f(y),f(y0)) = d(f(y),f(U˜z)). Then

d(f(y),f(˜y )) d(y, y˜ ) z ⌘ z ⌘(1). d(f(y),f(y ))  d(y, y )  0 ✓ 0 ◆

1 A similar computation applied to f will give that

1 d(x0, U˜z) d(x0, x˜z), C1 ·

where C1 depends only on ⌘. Now by condition (?) we have C2 := C(d(x0,y)) so that

1 d(x0, U˜z) d(y, U˜z) C2 d(x0, U˜z), C2 ·   ·

and condition (?) applied to f(x0) and f(y) gives C3 := C˜(d(f(x0),f(y))) so that

1 d(f(x0),f(U˜z)) d(f(y),f(U˜z)) C3 d(f(x0),f(U˜z)). C3 ·   ·

Hence: d(f(y),f(z)) d(f(y),f(U˜ )) ⌘(1)2 z d(y, z)  · d(y, U˜z) ˜ 2 d(f(x0),f(Uz)) ⌘(1) C2C3  · d(x0, U˜z)

2 d(f(x0),f(˜xz)) ⌘(1) C1C2C3  · d(x0, x˜z)

3 d(f(x0),f(xz)) ⌘(1) C1C2C3  · d(x0,xz) 2⌘(1)3C C C L (x ).  1 2 3 f 0 62 3 In particular this argument implies that Lf (y) 2⌘(1) C1C2C3Lf (x0). The fact that x0 is |B  |B f-good guarantees L (x ) < . Furthermore, this argument works for any p B. Since C ,C f 0 1 2 2 3 depend on the distance between y and x0, set

C =sup C(d(p, x )) : p B ,C =sup C˜(d(f(p),f(x )) : p B . 4 { 0 2 } 5 { 0 2 }

3 This gives that Lf (p) 2⌘(1) C1C4C5Lf (x0) for any p B. Since B is open there exists an |B  2 r > 0 so that B (y) B. For any x ,x B (y), there is a geodesic (in U) joining x ,x . y ry ⇢ 1 2 2 ry/3 1 2

Note that if x3 is any point on this geodesic, then by the triangle inequality,

2r r d(x ,y) d(x ,x )+d(x ,y) d(x ,x )+d(x ,y) < y + y = r . 3  1 3 1  1 2 1 3 3 y

In particular the geodesic joining x1 and x2 is contained in Bry (y). Furthermore, we have our estimate on L (x ) for any x [x ,x ]. Applying the argument in Lemma 5.1.9 gives f 3 3 2 1 2

d(f(x ),f(x )) 4⌘(1)3C C C L (x ) d(x ,x ) 1 2  1 4 5 f 0 · 1 2

for all x1,x2 Br /3(y). In particular, this means that f B (y) is Lipschitz. We repeat a similar 2 y | ry/3 1 1 argument for f to obtain a lower bound. Note that the relationships L (x )= and f 0 l 1 (f(x0)) f 1 1 l (x )= imply that f(x ) is K-good with respect to f . So, there is some r > 0 f 0 L 1 (f(x0)) 0 f(y) f 1 in which f B (f(y)) is Lipschitz. Shrink Br /3(y) to Br˜ (y) so that f(Br˜ (y)) Br (f(y)). | rf(y) y y y ⇢ f(y) Set B˜ = B (y). For a, b f(B˜): r˜y 2

1 1 d(f (a),f (b)) 3 4⌘0(1) C˜ C˜ C˜ L 1 (f(x )) d(a, b)  1 4 5 f 0

1 1 where C˜1 is obtained similarly to C1 and depends only on ⌘. Setting f (a)=p, f (b)=q, we have

d(p, q) 3 4⌘0(1) C˜ C˜ C˜ L 1 (f(x )). d(f(p),f(q))  1 4 5 f 0 63 This gives us the lower bound

d(f(p),f(q)) 1 . 3 d(p, q) ˜ ˜ ˜ 1 4⌘0(1) C1C4C5Lf (f(x0))

Set

3 3 C := max 4⌘(1) C C C , 4⌘0(1) C˜ C˜ C˜ . 6 { 1 4 5 1 4 5}

Using the fact that 1 = l (x ), we now have L 1 (f(x0)) f 0 f

1 d(f(p),f(q)) lf (x0) C6Lf (x0) KC6lf (x0) C6  d(p, q)  

for all p, q B˜. Hence f ˜ is biLipschitz (a quasi-similarity). 2 |B

So, Lemma 5.1.11 gives that f is locally biLipschitz when f is restricted to a fiber. One may consider an open ball B – on each fiber in the open ball you can ostensibly repeat the argument of Lemma 5.1.11 to get f being biLipschitz in the intersection of B with each fiber. The problem is that Lemma 5.1.11 gives a Lipschitz constant that depends on the K-good point for f of comparison (or rather, both K and lf (x0)). We need some sort of assurance that one is able to compare lf across fibers. The next lemma is our answer to this particular dilemma:

Lemma 5.1.12. Let y U and x0 U be a ﬁxed K-good point with respect to f. Let ">0 be 2 2 such that B (y) U does not contain x . For any x B (y) denote by U the ﬁber containing x. " \ 0 2 " x Let x0 U be a K0-good point with respect to f so that d(x, x0 ) d(y, x ). Then there exists 0 2 x 0 ⇡ 0 A 1 so that 1 l (x ) l (x0 ) Al (x ). A f 0  f 0  f 0

A is quantitative in ⌘, diam B (y) U,diam f(B ) f(U) and the f-good constants K and K0. " \ " \

Proof. Let x B (y). Now by Lemma 5.1.11 we have that f is a (M ,l (x ))-quasi-similarity. 2 " 1 f 0

Since the set of good points on Ux is dense we may apply the same argument in Lemma 5.1.11 to 64 get that 1 d(f(x),f(x0)) lf (x0) M2Lf (x0) M2  d(x, x0)  for all x0 on U near x. Furthermore, M depends on ⌘,K0, and the diameter of B (y) U x 2 " \ x and f(B (y) U ). Now let x0 U and y0 U such that d(x, x0)=d(y, y0)=d(x, y). By " \ x 2 x 2 quasi-symmetry we have 1 d(f(x),f(y)) ⌘(1) ⌘(1)  d(f(x),f(x0))  and 1 d(f(x),f(y)) ⌘(1). ⌘(1)  d(f(y),f(y0)) 

2 In particular this implies d(f(x),f(x0)) ⌘(1) d(f(y),f(y0)). Dividing both sides by d(x, y) we  get d(f(x),f(x )) d(f(y),f(y )) 0 ⌘(1)2 0 . d(x, x0)  d(y, y0)

This implies

1 2 lf (x0) ⌘(1) M1Lf (x0), M2 

2 so l (x0 ) ⌘(1) M M L (x ). Applying this argument to the other bounds we obtain f 0  1 2 f 0

1 0 2 lf (x0) Lf (x0). ⌘(1) M1M2 

So, we have:

1 2 0 0 0 0 2 lf (x0) Lf (x0) K lf (x0) K K⌘(1) M1M2lf (x0). ⌘(1) M1M2   

Since the assumption of the ambient space X is that the dense set of points is uniformly f-good, we may choose a uniform upper and lower bound.

We now give the proof of Theorem 5.1.7, which follows cleanly from the previous lemmas.

Proof of Theorem 5.1.7. Let y X and " > 0 be such that B (y) E. We show that f is a 2 1 "1 ⇢ 65 quasi-similarity in B(y) for some 0 <<"1. Choose K so that every good point in B"1 (y) is

K-good with respect to f. Let x0 be a K-good point with respect to f in B"1 (y); let 0 <"2 <"1 be so that f is a (M ,l (x ))-quasi-similarity in B (y) U. Let p, q B (y). Denote by U the 1 f 0 "2 \ 2 "2 p ﬁber of X containing p. Let p0 U be such that d(p, p0)=d(p, q) and p be a K-good point with 2 p 0 respect to f so that d(p, p ) d(y, x ). We also may have that f is a (M ,l (p ))-quasi-similarity 0 ⇡ 0 2 f 0 in B (y) U . By quasi-symmetry we have that "2 \ p

1 d(f(p),f(q)) ⌘(1). ⌘(1)  d(f(p),f(p0)) 

Rearranging and dividing all sides by d(p, q) gives

1 d(f(p),f(p )) d(f(p),f(q)) d(f(p),f(p )) 0 ⌘(1) 0 . ⌘(1) d(p, p0)  d(p, q)  d(p, p0)

Now by Lemma 5.1.12 we get in particular

d(f(p),f(q)) ⌘(1)M Kl (p ) ⌘(1)M KAl (x ) d(p, q)  2 f 0  2 f 0

Note that A depends on M and M , but these quantities really depend on the quantities sup d(x ,z): 1 2 { 0 z B (y) U and sup d(p ,z):z B (y) U , respectively, but these quantities are bounded 2 "2 \ } { 0 2 "2 \ p} in B"2 (y), as C is continuous. Hence, we may choose 0 <"3 <"2 so that we have a uniform upper bound in B"3 (y). We apply a similar argument to get 0 <<"3 so that we have a lower bound in

B(y).

Theorem 5.1.7 was intended to be applied to the theory of Carnot groups – in particular, the Heisenberg group. However, it turns out that condition (?) is not satisfied in the Heisenberg group . Consider as the Lie algebra X Y Z with X = e ,Y = e ,Z = e , and [e ,e ]=e . H H h 1i h 2i h 3i 1 2 3 We take as fibers of the left cosets of the subgroup corresponding to X, in other words, left- H translates of the x-axis. Now, the distance from 0 X to the fiber (ye +ze ) X is comparable to y + z 1/2. However, 2 2 3 ⇤ | | | | 66 for te X, we see that d(te , (ye + ze ) X) is comparable to the quantity y + z yt 1/2 1 2 1 2 3 ⇤ | | | | which is at most y + z 1/2 + yt 1/2. In the special case that z =0, | | | | | |

d(0, (ye ) X) y ,d(te , (ye ) X) y + yt 1/2. 2 ⇤ ⇡| | 1 2 ⇤ ⇡| | | |

So, moving t away from the origin along the x-axis increases distance from (ye ) X quickly if 2 ⇤ y and t are small. In particular, Condition (?) is not satisﬁed. | | | | 5.2 Rigidity of Quasiconformal Maps on Two-Step Carnot Groups

In Le Donne and Xie (2016) the following rigidity theorem is proven:

Theorem 5.2.1. (Le Donne and Xie, 2016) Let G be a Carnot group with reducible ﬁrst stratum. Then every quasisymmetric map F : G G is a quasi-similarity, quantitatively. !

This theorem illustrates the following phenomenon: if f : G G is a globally defined qua- ! siconformal mapping, with G a Carnot group, then often f will be biLipschitz. In choosing a dissertation project, the goal was to somehow generalize this result by removing the restriction that f must be globally defined, and changing it to be defined merely on an open subset of the group G. This idea is manifested in the following theorem:

Theorem 5.2.2. Let G = V1 V2 be a two-step Carnot group and let W1 V1 be a subspace and ✓ set W := [W ,W ]. Let W = W W ; W is a Lie sub-algebra of g and so corresponds to a Lie 2 1 1 1 2 subgroup. Suppose that E G is an open set. If f : E G is a quasiconformal mapping that ✓ ! permutes the left cosets of W , then f is locally biLipschitz.

The proof of Theorem 5.2.2 proceeds much like the proof of Theorem 5.1.7; ﬁrst, we shall prove some technical lemmas that help with the proof of the main result.

Lemma 5.2.3. Let G, V1,V2,W,W1,W2 be as above. Then divergence of ﬁbers is at worst quadratic: that is, for p, q g W , and h W is any other ﬁber so that d(p, h W ) << 1, we have 2 ⇤ ⇤ ⇤

d(p, h W ) C d(q, h W )1/2, ⇤  · ⇤ 67 where C depends on the distance between p and q.

Proof. This is, in some sense, a consequence of the BCH formula. We ﬁrst compute the distance between a point in W to a point in a coset of W . Deﬁne W1? and W2? to be the orthogonal complements of W ,W in V ,V , respectively. Let w + w W with w W ,w W . 1 2 1 2 1 2 2 1 2 1 2 2 2

Denote by ⇡2 the projection of a point in V2 onto W2 and ⇡2? the projection of a point in V2 onto

W ?. In general, a ﬁber g W has the form (x + y) W , where x W ? and y W ?. So, a point 2 ⇤ ⇤ 2 1 2 2 in g W is given by (x + y) (˜w +˜w ), with w˜ W and w˜ W . We have, by 4.2.4: ⇤ ⇤ 1 2 1 2 1 2 2 2

1 ( w w ) (x + y) (˜w +˜w )= ( w + x)+( w + y) [w ,x] (˜w +˜w ) 1 2 ⇤ ⇤ 1 2 1 2 2 1 ⇤ 1 2 ✓ ◆ 1 1 =(˜w w + x)+ w˜ w [w , w˜ ]+ ⇡ [x, w +˜w ] 1 1 2 2 2 1 1 2 2 1 1 ✓ ◆ 1 + y + ⇡?[x, w +˜w ] . 2 2 1 1 ✓ ◆

Simplifying this expression gives

1 1 1 (˜w w + x) + w˜ w [w , w˜ ]+ ⇡ [x, w˜ + w ] + y + ⇡?[x, w˜ + w ] . 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 V1 ✓ ◆ ✓ ◆ 2 W2 W 2 2? | {z } 2 | {z } | {z } Hence the distance of w + w to (x + y) W is the minimum of the norm of the above expression 1 2 ⇤ over all w˜ +˜w W . Note that picking w˜ = w and w˜ = w ⇡ [x, w ] simpliﬁes the 1 2 2 1 1 2 2 2 1 expression to

x + y + ⇡2?[x, w1], and the norm of this element is comparable to the minimum. To see this fact, we ﬁrst note that there exists K so that for all X, Y g, [X, Y ] K X Y . Furthermore, the norm of ( w 2 | | | || | 1 w ) (x + y) (˜w +˜w ) is comparable (up to a multiplicative constant) to 2 ⇤ ⇤ 1 2

1 1 1/2 1 1/2 w˜ w +x + w˜ w [w , w˜ ]+ ⇡ [x, w˜ + w ] + y + ⇡?[x, w˜ + w ] . (5.2.4) | 1 1 | 2 2 2 1 1 2 2 1 1 2 2 1 1 68 1/2 Let a = y + ⇡?[x, w ] . We must then show that the expression in (5.2.4) is greater than or | 2 1 | equal to C( x + a) | | for some appropriately chosen constant C, since x + a is (5.2.4) with the choice w˜ = w and | | 1 1 w˜ = w ⇡ [x, w ]. We have two cases: 2 2 2 1

Case 1. w w˜ 1 a. | 1 1| 100 Here, we are essentially done. This is because the expression in (5.2.4) is greater than or equal to

1 1 1 w˜ w + x ( w˜ w + x ) a + x . | 1 1 |10 | 1 1| | | 10 100 | | ✓ ◆

Case 2. w w˜ 1 a. | 1 1| 100 This can be broken down into two subcases:

Case 2.1. x 1 a. | | 100K First we note that w w˜ + x x since x W ? and w w˜ W . Furthermore, | 1 1 || | 2 1 1 1 2 1

1 1/2 1 1/2 x + y + ⇡?[x, w +˜w ] = x + y + ⇡?[x, w + w w +˜w ] | | 2 2 1 1 | | 2 2 1 1 1 1 1/2 1 = x + y + ⇡?[x, w1]+ ⇡?[x, w˜1 w1] | | 2 2 2 1/2 1 = x + y + ⇡?[x, w1] ⇡?[x, w1 w˜1] . | | 2 2 2 By the reverse triangle inequality,

1 1/2 1 1/2 x + y + ⇡?[x, w ] ⇡?[x, w w˜ ] x + a ⇡?[x, w w˜ ] | | 2 1 2 2 1 1 | | 2 2 1 1 1 x + a a | | 100p2 100p2 1 = x + a. | | 100p2

Case 2.2. x 1 a. | | 100K We have that x + w w˜ x , and so the expression in (5.2.4) is at least | 1 1|| |

101K K 100K x K a x = x = x + | | x + . | | 101K | | 101K | | 101K 101K | | 101K

Hence, the norm of the expression in (5.2.4) is comparable (up to a multiplicative constant depending on K) to x + a. We now consider two elements p, q of g in the same ﬁber and consider | | their distance from an arbitrary ﬁber (x + y) W . For ease of computation we assume one of these ⇤ elements is the identity in g (since the distance is left-invariant). Let p = w1 + w2. We again have two cases:

Case 1. ⇡2?[x, W1]=0. In this case, d(p, (x + y) W ) x + y 1/2, so d(p, (x + y) W ) d(0, (x + y) W ). In ⇤ ⇡|| | | ⇤ ⇡ ⇤ 70 particular, d(p, (x + y) W ) C d(0, (x + y) W )1/2 for some constant C. ⇤  · ⇤

Case 2. ⇡?[x, W ] =0. 2 1 6 We have:

1/2 d(w + w , (x + y) W ) x + y + ⇡?[x, w ] 1 2 ⇤ ⇡| | | 2 1 | 1/2 1/2 x + y + ⇡?[x, w ] | | | | | 2 1 | x + y 1/2 + pK x 1/2 w 1/2 | | | | | | | 1| x + y 1/2 + pK x 1/2 w 1/2 + w K y 1/4 | | | | | | | 1| | 1| | | (1 + w K)( x + y 1/2)1/2, p  | 1| | | | | p where K is such that [X, Y ] K X Y for all X, Y g. We note that w depends on the | | | || | 2 | 1| point p but since w is the distance of w to the identity, this constant depends on the distance | 1| 1 from 0 of the point w + w , as desired. Furthermore if y =0then d(0, (x +0) W ) x 1 2 ⇤ ⇡|| and d(w + w , (x +0) W ) x + pK w 1/2 x 1/2, so this estimate is achievable for certain 1 2 ⇤ ⇡| | | 1| | | ﬁbers.

Lemma 5.2.5. Let E G be open and f : E G be an ⌘-quasisymmetry permuting the cosets ⇢ ! of W . Suppose B is a bounded subset of W and let x B be a K-good point with respect to f. 2 Then L (x) M, where M depends on the diameter of B. f 

Proof. Let p B be another K-good point in B with respect to f. Earlier, we concluded that there 2 was a “worst case” as far as divergence of ﬁbers: the quantity d(x, g W ) is at worst C [d(p, g ⇤ 1 · ⇤ 1/2 W )] , where C1 depends on the distance between x and p. In fact, we may choose a sequence of

ﬁbers (gi W ) converging to W so that this estimate is achieved for every i N (see the proof of ⇤ 2 1/2 Lemma 5.2.3). Set r = d(p, g W ); then we have that d(x, g W ) C r . Let x0 g W i i ⇤ i ⇤ ⇡ 1 · i i 2 i ⇤ be such that

d(f(x),f(g W )) = d(f(x),f(x0 )). i ⇤ i 71

Now since the ﬁbers gi W converge to W , eventually (i N for some N N) we will have ⇤ 2

d(f(x),f(g W )) d(f(p),f(g W )) i ⇤ L (x), i ⇤ L (p). d(x, g W ) ⇡ f d(p, g W ) ⇡ f i ⇤ i ⇤

To see this, we let p g W be such that d(p, g W )=d(p, p ). Now as p p, f(p ) f(p) i 2 i ⇤ i ⇤ i i ! i ! and so eventually we will have

d(f(p),f(gi W )) d(f(p),f(pi0 )) ⇤ 2Lf (p). d(p, g W )  d(p, p0 )  i ⇤ i

Naturally this inequality fails if Lf (p)=0but we recall that p is a K-good point with respect to f, hence L (p) > 0. Similarly if p˜ g W is such that d(f(p),f(g W )) = d(f(p),f(˜p )), then f i 2 i ⇤ i ⇤ i as i , p˜ p and so we will eventually have !1 i !

d(f(p),f(g W )) d(f(p),f(˜p )) 1 1 i ⇤ i l (p) L (p). d(p, g W ) d(p, p˜ ) 2 f 2K f i ⇤ i

A similar argument applies with L (x). Let a be such that L (x)=a L (p). Let s = f f · f i d(f(p),f(g W )). Then we have, for i sufﬁciently large, i ⇤

s d(f(p),f(p0 )) 2L (p) d(p, p0 ). (5.2.6) i  i  f · i

Hence, we have:

d(f(x),f(x0 )) L (x) d(x, x0 ) i ⇡ f · i L (x) d(x, g W ) f · i ⇤ = aL (p) d(x, g W ) f · i ⇤ aL (p) C r1/2 ⇡ f · 1 · i 1 1/2 aLf (p) C1 si (from 5.2.6) · · Lf (p) · = a L (p) C ps1/2. f · 1 · i q 72 So,

1/2 d(f(x),f(x0 )) K a L (p) C s (5.2.7) i · · f · 1 · i q for some constant K 1. Now the “worst case” bound of divergence also applies to the ﬁbers f(W ) and f(g W ), so we have i ⇤

1/2 d(f(x),f(g W )) = d(f(x),f(x0 ) C s , (5.2.8) i ⇤ i  2 · i

where C2 depends on the distance between f(x) and f(p). Combining inequalities 5.2.7 and 5.2.8 yields a K C L (p) C . (5.2.9) · · 1 · f  2 q Now, we have that L (x) a = f . Lf (p)

Applying this relationship to 5.2.9 gives

C2 Lf (x) Lf (x) a Lf (p)= Lf (p)= . (5.2.10) K C1 · Lf (p) L (p) · q q f p Inequality 5.2.10 can be summarized as

L (x) C L (p). (5.2.11) f  · f q

One may run through the entire argument again with the roles of p and x switched; this gives

L (p) C˜ L (x). f  f q

Combining this with 5.2.11 establishes the inequality

L (x) C L (p) C C˜ 4 L (x), f  · f  · · f q p q 73 and hence we ﬁnally have L (x)3/4 C C.˜ f  · p Now C and C˜ depend on the distance between x, p and f(x),f(p), respectively, so we can take p the supremum of this quantity over B to get an upper bound for Lf (x) (which consequently works for any x B). 2

Proof of Theorem 5.2.2. Let E G be open with f : E G quasiconformal. Let y U. Since ⇢ ! 2 the metric is translation-invariant it suffices to show the result in the case that y W . There 2 exists a "1 > 0 so that f is ⌘-quasisymmetric in B"1 (y) for some ⌘. Now by Lemma 5.2.5, there exists an M>0 so that L (x) M for all f-good points x B (y); set " = " /2 and let f  2 "1 2 1 x (B (y) B (y)) W be a K-good point with respect to f. Note that while we do not have 2 "1 \ "2 \ condition (?), we may find fibers g W so that ⇤

1 1/2 1/2 d(x, g W ) d(y, g W ) C1 d(x, g W ) . C1 ⇤  ⇤  · ⇤

We may do this because we may select ﬁbers g W so that our worst-case estimate for divergence ⇤ is achieved; here C1 depends on the distance between x and y. So, we may repeat the argument of Lemma 5.1.11 to obtain that, for z B (y) W , 2 "2 \

1/2 Lf (z) C2M , |W 

where C2 depends on the distance between x and z, but we may take the supremum of these constants over all z B (y) W ; in particular, this gives that f is biLipschitz in B (y) W 2 "2 \ "2 \ by Lemma 5.1.9. Now if p is a point in B (y),p W is the coset of W containing p. Let x0 be "2 ⇤ an f-good point on p W B (y). Since diam p W B (y) diam B (y), L (x0) M as ⇤ \ "1 ⇤ \ "1  "1 f  well, and so we reapply the argument in 5.1.11 to get a bound on Lf (p) depending on the distance between p and x0. Then, we simply take the supremum of the Lf (p) and get a uniform bound on L (p) for all p B (y). It then follows by the proof of 5.1.9 that f is biLipschitz in B (y). f 2 "2 "2 74

We note that the proof above hinges on the fact that divergence of ﬁbers in G is quadratic, that is, if x, y W , then 2 d(x, g W ) C d(y, g W )1/2. ⇤  · ⇤

In the general k-step case this will no longer be true and so generalizing the result will require a somewhat different technique. Nevertheless, it is the author’s hope to successfully generalize Theorem 5.2.2 to k-step Carnot groups. As an application of the theorem, we ﬁrst introduce the following deﬁnition:

Definition 5.2.12. (Le Donne and Xie, 2016) Let G be a Carnot group and let g = V1 Vk ··· be a stratification of the Lie algebra g of G. Recall that a Lie algebra isomorphism f : g g is ! strata-preserving if f(Vi)=Vi for 1 i k. We say that the first stratum of G is reducible, or   that G has reducible first stratum, if there exists a non-trivial proper subspace W1 of V1 so that f(W1)=W1 for all strata-preserving Lie algebra isomorphisms f.

Then, we have the following application of 5.2.2:

Corollary 5.2.13. Let G be a 2-step Carnot group with reducible ﬁrst stratum. Let E G be ⇢ open. Then any quasiconformal mapping f : E G is locally biLipschitz. !

This follows from Theorem 5.2.2 once we know that f permutes the cosets of W (checking the conditions necessary to make G into a ﬁbered metric space are addressed in Le Donne and Xie (2016) in section 4), but any quasiconformal mapping on a Carnot group with reducible ﬁrst stratum has this property, thanks to this fact of Xie (2013):

Proposition 5.2.14. Let G be a Carnot group with Lie algebra g = V1 Vk. Let W1 V1 ··· ⇢ be a subspace and denote by W the Lie subalgebra of g generated by W1. Let H be the connected Lie subgroup of G corresponding to W . Let f : G G be a quasisymmetric homeomorphism. If ! the Pansu differential dF satisfies (dF )(x)(W ) W for almost all x G, then f maps left cosets ⇢ 2 of H into left cosets of H. 75 There are many Carnot groups with reducible first stratum. For example, every Carnot algebra is the subalgebra of some Carnot algebra with reducible first stratum – one may see section 4 in the paper by Xie (2017) for the general construction and section 6 for specific examples. Recall that quasi-isometries of hyperbolic spaces induce quasiconformal mappings on the Gro- mov boundary. Conversely, quasiconformal mappings on the Gromov boundary may be extended to quasi-isometries of the underlying space. Since Carnot groups arise as the boundary of negatively curved Lie groups, the hope is that being able to make statements about quasiconformal mappings on Carnot groups says something about some underlying quasi-isometric rigidity properties of these negatively curved Lie groups. 76

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