On the Rate of Convergence to the Asymptotic Cone for Nilpotent Groups and Subfinsler Geometry

On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry

Emmanuel Breuillard1 and Enrico Le Donne

Laboratoire de Mathématiques, Université Paris Sud 11, 91405 Orsay, France

Edited by Gregory A. Margulis, Yale University, New Haven, CT, and approved October 30, 2012 (received for review April 13, 2012)

Addressing a question of Gromov, we give a rate in Pansu’s the- asymptotic cones and yet are not ð1; CÞ-quasi-isometric for any orem about the convergence to the asymptotic cone of a finitely constant C > 0, answering negatively a related question raised by generated nilpotent group equipped with a left-invariant word Burago and Margulis (6). 1 metric rescaled by a factor n. We obtain a convergence rate (mea- For general nilpotent groups, we obtain a rate of convergence − 2 2 1 – n 3r − − sured in the Gromov Hausdorff metric) of for nilpotent in O n 3r if r > 2, and O n 2 if r = 2, where r is the nilpotency −1 groups of class r > 2 and n 2 for nilpotent groups of class 2. We class of Γ. The case r = 2 is a simple consequence of a result of also show that the latter result is sharp, and we make a connection Stoll (7), and we use it to obtain the 2 exponent for general r. between this sharpness and the presence of so-called abnormal 3r The above example on H ðZÞ × Z shows that this error term is geodesics in the asymptotic cone. As a corollary, we get an error 3 d d− 2 sharp for groups of nilpotency class 2. However, it is likely not to term of the form vol(B(n)) = cn + O(n 3r ) for the volume of Cay- r be sharp for groups of nilpotency class 3 or more, and we con- ley balls of a general nilpotent group of class . We also state 1 a number of related conjectural statements. jecture that the exponent 2 holds for all nilpotent groups and is therefore independent of r. – polynomial growth | subRiemannian geometry The rate of convergence in Gromov Hausdorff metric toward the asymptotic cone implies a rate of convergence in the volume asymptotics of Cayley balls. In particular, we obtain as a corollary n his fundamental paper on groups with polynomial growth, that, for every nilpotent group Γ with generating set S (S finite, Gromov (1) observed that Cayley graphs of finitely generated − I S = S 1 ∈ S groups with polynomial growth, when viewed from afar, admit , and 1 ), we have that limits that are Lie groups endowed with a certain left-invariant −β n = d + d r ; → + ∞; [1] geodesic metric. This was a simple but basic step in his proof that jS j cSn OS n as n groups with polynomial growth are virtually nilpotent (i.e., have fi β = 2 > n a nilpotent subgroup of nite index). where r 3r for r 2 and S is the ball of radius n in the word In his thesis, Pansu (2) established that, if we start with the metric and cS > 0 is a constant. Cayley graph of a nilpotent group, then there is a unique limit. In Stoll had showed in (7) that one can take β = 1 for groups 2 other words, the sequence of metric spaces Cay Γ; 1 ρ of of nilpotency class at most 2, but nothing seemed known for n S n∈N higher-step groups, even though it is a folklore conjecture that Γ scaled down Cayley graphs of the nilpotent group with gen- β = 1 should hold for all r. We also note that the very fact that ρ – r erating set S and word metric S converges in the Gromov an error term exists at all is also a distinctive feature of nilpotent Hausdorff topology (3) toward a certain explicit left-invariant groups. Indeed, there is a class of (nondiscrete) groups of poly- subFinsler metric on a nilpotent Lie group: the Pansu limit 2 nomial growth (of the form R ⋊ θZ, where Z acts by an irrational metric on the asymptotic cone of Γ (Fig. 1). rotation with angle θ), which are solvable but not nilpotent, The goal of this note is to study the rate of convergence in ’ for which it can be shown that the volume asymptotics (here Pansu s theorem and give quantitative estimates. This question ≅ 3 θ was posed by Gromov [ref. 4, §2C remark 2C2(a)]. It requires volðBðnÞÞ cn ) admit no error term whatsoever if is chosen approximating (with explicit bounds) word metric geodesics in Γ carefully (8). with subFinsler geodesics in the Lie group that is the asymptotic In this note, we present the aforementioned results and give cone of Γ and vice versa. This problem is intimately connected some information on their proofs, although full details will be ∼ to the underlying geometry of nilpotent Lie groups endowed with given in a separate text (www.math.u-psud.fr/ breuilla/Balls.pdf) left-invariant subFinsler metrics. due to lack of space. One of our key findings is that the quality of the error term in The note is organized as follows. First, we state our main result Pansu’s theorem is related to the presence or the absence of so- about the rate of convergence to the asymptotic cone (Theorem called abnormal geodesics in the asymptotic cone. These geo- 2) and explain the strategy of the proof and its main ingredients. desics do not exist in classical Riemannian or Finsler geometry, In the second part of the paper, we present the example showing but are typical in subRiemannian or subFinsler geometry (5). We 1 the sharpness of the exponent 2 for step-2 groups (Theorem 7 and show that their presence worsens the error term in the conver- Proposition 8). We then describe some ideas in the proof of the gence to the asymptotic cone for general nilpotent groups. For sharpness result, in particular a detailed study of the geometry of example, if Γ = Zn, the free Abelian group of rank n,orif Γ = Z + subFinsler metrics on the Heisenberg group and its direct H2n+1ð Þ, the 2n 1-dimensional Heisenberg group, equip- product with R. Finally, in the last part of the note, we prove the ped with a word metric, then the asymptotic cone of Γ bears no volume estimate ½1, discuss the volume asymptotics conjecture, and abnormal geodesics, and it can be shown that the convergence to its relation to some well-known conjectures in subRiemannian the asymptotic cone is best possible, namely with a rate 1.On n geometry. the other hand, if Γ = H3ðZÞ × Z, the speed of convergence toward the asymptotic cone depends on the word metric, and while 1 for some generating sets the speed may be optimal, i.e., in n,we show that, for some choices of generating sets, the rate of con- Author contributions: E.B. and E.L.D. wrote the paper. −1 vergence is no faster that n 2. This result is due to the fact that The authors declare no conflict of interest. the asymptotic cone H3ðRÞ × R admits abnormal geodesics in the This article is a PNAS Direct Submission. direction of the second R factor (Figs. 2 and 3). In particular, we 1To whom correspondence should be addressed. E-mail: emmanuel.breuillard@math. exhibit two word metrics on H3ðZÞ × Z that have isometric u-psud.fr.

19220–19226 | PNAS | November 26, 2013 | vol. 110 | no. 48 www.pnas.org/cgi/doi/10.1073/pnas.1203854109 Downloaded by guest on September 25, 2021 measured using Pansu limit norm ║ · ║∞. A piecewise horizontal SPECIAL FEATURE path is by definition the concatenation of finitely many segments of one parameter subgroups of G∞ of the form fexpðtXÞgt∈½0;T for some X ∈ g=½g; g. The distance d∞ is geodesic and left-invariant by definition. We call it subFinsler, because the norm ║ · ║∞ is not a Euclidean norm but a polyhedral norm. It is not a Finsler metric however (if g is non-Abelian), because the norm is only defined on a subspace of the Lie algebra. It can be checked that this subspace g=½g; g z generates the whole Lie algebra. This implies [Chow’stheorem (5)] that every two points in G∞ can be joined by a piecewise fi fi y linear horizontal path and thus d∞ is well-de ned. We note nally x that it can be shown (10) that any left-invariant geodesic metric on a connected Lie group is a subFinsler metric for some norm on a generating subspace of the Lie algebra. We call the distance d∞ the Pansu limit metric of ðΓ; SÞ.Pansu showed in his thesis that ðG∞; d∞Þ is the asymptotic cone of ðΓ; ρ Þ. More precisely he showed that the sequence of renormal- S Γ; 1 ρ Γ ; ized Cayley graphs Cay n S of converges toward ðG∞ d∞Þ in the Gromov–Hausdorff topology on pointed metric spaces (based ρ Γ fi at id). Here S is the left-invariant word metric on de ned ρ ; := ∈ N; −1 ∈ n by Sðx yÞ inffn x y S g. > := ; ; ; For any R 0, set X∞ðRÞ ðBd∞ ðid RÞ d∞Þ, where Bd∞ ðid RÞ is the closed ball of radius R in ðG∞; d∞Þ. Similarly, we set Fig. 1. The unit ball for the Pansu limit metric d3 of the Cayley graph of the Z discrete Heisenberg group H3ð Þ with standard generators (for an explicit := ; ; 1 ρ ; Xn RÞ BSðid RnÞ n S , where BSðid RnÞ is the closed ball of formula for d3 see Fine Geometry of the Heisenberg Group Equipped with

Γ; ρ MATHEMATICS the Pansu Metric below). radius Rn in the Cayley graph ð SÞ. Then

Theorem 1 [Pansu (2)]. For every R > 0, XnðRÞ converges to X∞ðRÞ in Rate of Convergence to the Asymptotic Cone the Gromov–Hausdorff topology. We now recall Pansu’s description of the asymptotic cone of Recall that the Gromov–Hausdorff metric between any two Γ a nilpotent group and state our main theorem. Let be a tor- compact metric spaces ðX; dX Þ and ðY; dY Þ is defined as sion-free nilpotent group generated by a finite set S (we assume ∈ −1 = Γ 1 S and S S). It is well-known (9) that embeds as a ; := ; ; = ⊔ ; ; cocompact discrete subgroup of a simply connected nilpotent dGH X Y inf dHaus;Z X Y Z X Y dZ admissible Lie group G, its Malcev closure. Let g be the Lie algebra of G. One can associate to g another nilpotent Lie algebra, called the where an admissible metric dZ on the disjoint union Z = X ⊔ Y is graded Lie algebra of g and denoted by g∞ by setting a distance on Z which coincides with dX on X and with dY on Y. ðiÞ ði+1Þ Here is our main result: g∞ = ⊕i≥1g =g ; Theorem 2. There is a positive constant αr > 0 depending only on the where gði+1Þ = ½g; gðiÞ is the central descending series of g and where nilpotency class r of Γ such that, as n → + ∞, the Lie bracket is defined in the natural way by the formula ½x mod i+1 j+1 i+j+1 i j ð Þ; ð Þ = ; ð Þ ∈ ð Þ ∈ ð Þ −αr g y mod g ½x y mod g for x g and y g . dGH ðXnð1Þ; X∞ð1ÞÞ = OSðn Þ: The exponential map establishes a diffeomorphism between g and the Lie group G and between g∞ and a simply connected Lie α = α = 1 α = 2 > group G∞. We denote the group law on G∞ by x p y to distin- Moreover one can take 1 1, 2 2, and r 3r if r 2. guish it from the group law on G, for which we simply write x · y. Note that Xnð1Þ as a set is the ball of radius n for the word The projection Lie algebra homomorphism ρ Γ metric S on . Note also that by scaling, this also implies that π : → = ; [2] g g ½g g 1−αr −αr dGH ðXnðRÞ; X∞ðRÞÞ = OS R n lifts to a group homomorphism π : G → g=½g; g which, by a slight π π Γ abuse of notation, we also denote by . The image ð Þ is a dis- > crete cocompact subgroup of the vector space g=½g; g generated for every R 0. π π fi = ; Theorem 2 addresses a question of Gromov [ref. 4, Remark by ðSÞ. In particular ðSÞ de nes a norm on g ½g g whose unit – π ║ · ║ 2C2(a)]. Speaking of the Gromov Hausdorff distance between ball is the convex hull of ðSÞ. We call this norm ∞ the Pansu X ð1Þ and X∞ð1Þ, he says “any bound on these distances would Γ; n limit norm of the pair ð SÞ. be a pleasure to have in our possession, even in the case of word = ; Viewing g ½g g as a subspace of the graded Lie algebra g∞, metrics on Γ” (4). fi we may de ne a left-invariant subFinsler metric on G∞ with Although we have proved Theorem 2 for word metrics on Γ only, horizontal subspace g=½g; g and norm ║ · ║∞. In other words, using Burago’s theorem [(11) or (12) for different proof], one can there is a left-invariant geodesic metric d∞ on G∞ defined by adapt our arguments and prove a similar result (at least with α ≥ 1=2r)forΓ-invariant Riemannian metrics on the Malcev clo- ; := γ ; r d∞ðx yÞ inffLð Þg sure G of Γ and more generally for Γ-invariant coarsely geodesic metrics on G. We will not pursue this here. where the infimum is taken over all piecewise linear horizontal Our result is sharp for r = 1; 2. The sharpness in case r = 2is paths γ from γð0Þ = x to γð1Þ = y, and LðγÞ is the length of γ discussed below and is related to a conjecture of Burago and

Breuillard and Le Donne PNAS | November 26, 2013 | vol. 110 | no. 48 | 19221 Downloaded by guest on September 25, 2021 Fig. 2. The section of the ball in the plane y = 0; note the cusps where the vertical direction is squashed.

;δ Margulis (6) and to the presence of abnormal geodesics in the ≠ dðid tðgÞÞ which given any g id, the ratio t oscillates between two asymptotic cone. However, we believe that the exponent is not distinct positive values as t → + ∞. Nevertheless if the metric dðx; yÞ > 1 ≥ sharp for r 2 and that the exponent 2 holds for all r 2. Proving is coarsely geodesic, then this ratio does converge and the analog of this would require a deeper understanding of the subFinsler geo- Pansu’s theorem holds [see (8, Theorem 1.4) for a proof]. desics of d∞ that we have so far. The proof of Theorem 2 is, to ’ The above proposition dealt only with G-invariant metrics. For a large extent, an effectivization of Pansu s proof, where we have to Theorem 2, one also needs to consider metrics on the discrete replace several compactness arguments used by Pansu by effective cocompact subgroup Γ, or more generally, Γ-invariant metrics on arguments with explicit bounds. The exponent 1 when r = 2isde- ’ 2 G. This is more involved, because small errors made in dis- duced from a theorem of Stoll (7), and Stoll s result is also used to cretizing at various places in the group can accumulate and 2 > obtain the exponent 3r for r 2. generate a large error in the end. Let G be a simply connected nilpotent Lie group and Γ adiscrete Comparison of Metrics on a Nilpotent Lie Group ρ cocompact subgroup of G.Let S be the left-invariant word metric To prove Theorem 2, one needs to first have some understanding on Γ associated to a finite symmetric set S. The following result is of the large-scale behavior of subFinsler metrics on nilpotent Lie very closely related to Theorem 2 and gives an estimate of the error groups. Given two left-invariant subFinsler metrics d and d on ρ 1 2 term between S and G-invariant subFinsler metrics on G. a simply connected nilpotent Lie group G, the following gives a simple criterion for when they are asymptotic and also gives an Theorem 4. Let d be a left-invariant subFinsler metric on G. The error estimate. following are equivalent: dðid;γÞ (i) ρ ;γ → 1, as γ → ∞ in Γ, Proposition 3. Given two left-invariant subFinsler metrics d1 and d2 Sðid Þ on G, the following are equivalent: (ii) the projection under π (see equation [2]) of the unit ball ; d1ðid gÞ → → ∞ of ║ · ║ istheconvexhullofπðSÞ,and (i) d ðid;gÞ 1, as g in G, d 2 −α ; γ − ρ ; γ = ; γ 1 r γ → ∞ Γ α (ii) the projection under π (see equation [2]) of the unit balls (iii) jdðid Þ Sðid Þj Oðdðid Þ Þ, as in , where r ║ · ║ ║ · ║ is as in Theorem 2. of 1 and 2 coincide , and −1 ; − ; = ; 1 r → ∞ α ≤ 1 (iii) jd1ðid gÞ d2ðid gÞj O d1ðid gÞ , as g in G. This is consistent with Proposition 3 because r r.One may wonder however if, given S and ρ , there is a distin- ║ · ║ ║ · ║ fi S The norms 1 and 2 are the norms used to de ne d1 and guished G-invariant subFinsler metric on G that best d2. respectively, as recalled in the last section. Similarly, one can approximates ρ . A good candidate for this is the following ; ; S prove that ðG d1Þ and ðG d2Þ have isometric asymptotic cones choice of subFinsler metric, which we call the Stoll metric = ; if and only if the normed vector space g ½g g endowed with the associated to ðΓ; SÞ. ║ · ║ projection of 1 is isometric to the same space endowed with Identifying G with its Lie algebra g via the exponential map, the projection of ║ · ║ . we may view the finite symmetric set S as a subset of g and take 2 ‘ Item (ii)inProposition 3 can also be replaced with the pro- its convex hull. It spans a vector subspace V of g. Let ║ · ║ be π ’ S S jection under of the unit balls of d1 and d2 coincide . In this the norm on VS whose unit ball is the convex hull of S. Then, form the proposition applies also to word metrics, instead of ║ · ║ induces a left-invariant subFinsler metric d on G, which subFinsler metrics, that is, the pseudodistances on G of the form S S −1 n we call the Stoll metric. dU ðx; yÞ = inffn ∈ N; x y ∈ U g, where U is a bounded symmetric neighborhood of id in G. Conjecture 5. There is a constant C = CðSÞ > 0 such that We remark here that it is crucial in the above proposition that the metrics be subFinsler or at least coarsely geodesic. Indeed, ρ ; γ − ; γ ≤ I. Babenko [see the last section of (13)] exhibited an example of j Sðid Þ dSðid Þj C a left-invariant distance dðx; yÞ on the Heisenberg group, endowed with the dilations δt (see the definition below ð3Þ), for for all γ ∈ Γ.

19222 | www.pnas.org/cgi/doi/10.1073/pnas.1203854109 Breuillard and Le Donne Downloaded by guest on September 25, 2021 The two parts of the proposition are fairly distinct. The first ρ ; γ ≥ SPECIAL FEATURE one follows immediately from Proposition 3, because Sðid Þ dSðid; γÞ for all γ ∈ Γ, where dS is the Stoll metric defined in the previous section. The second part of the proposition lies deeper ρ as we need to approximate a d∞-geodesic in G with a S-geodesic in Γ. This is done by first splitting a d∞-geodesic in G between id 1 and δnðxÞ into m ≅ n3 intervals of equal length so that δnðxÞ = y1 · ... · ym. Then one shows using the Campbell–Baker– Hausdorff formula combined with Lemma 2 below that y1 · ... · ym can be approximated by π2ðy1Þ · ... · π2ðymÞ with only n π ∈ = ð3Þ an error of order at most m2=r . Here, 2ðyiÞ G G is viewed as an element of G by viewing g=gð2Þ⊕gð2Þ=gð3Þ as a subspace of g ≅ g∞. Finally, one applies Lemma 1, which itself relies on Stoll’s result (7) to approximate each π2ðyiÞ by a suitable element of Γ and this ends the proof. · ... · π · ... · π Fig. 3. Half of the section of the ball, with the abnormal geodesic in red. The approximation of y1 ym by 2ðy1Þ 2ðymÞ as well as the proof of Proposition 3 relies mainly on the Campbell– Baker–Hausdorff formula and the following simple fact [also In a very elegant short paper (7), M. Stoll proved this claim* when used in (2)], which is itself a version of the classical Gronwall’s the nilpotency class of Γ is at most 2. It remains an open problem lemma. We note that a similar argument was used by the for r > 2. In fact, even the existence of some G-invariant metric on second-named author in (14, Appendix). ρ G that lies at a bounded distance from S seems unknown. Of course, by Proposition 3, the Stoll metric and the G-in- Lemma 2 (Gronwall-Type Lemma). Let G be a Lie group. Let ║ · ║ be variant Pansu distance (i.e., the Carnot-Caratheodory metric some norm on the Lie algebra of G, and let deð · ; · Þ be a Rie- induced by a choice of a complement of ½g; g in g that carries the mannian metric on G. Then, for every L > 0, there is a constant C = ; ║ · ║; > ξ ; ξ : ; → norm whose unit ball is the convex hull of the projection of S Cðde LÞ 0 with the following property. Assume 1 2 ½0 1 ξ = ξ = onto this complement) are asymptotic, but the Stoll metric seems G are two piecewise smooth paths in G with 1ð0Þ 2ð0Þ id. fi ξ′ ∈ MATHEMATICS to capture the ner behavior of the word metric. Let i LieðGÞ be the tangent vector pulled back at the identity = ξ′ ≤ The fact that Conjecture 5 holds for r 2 has the following by a left translation of G. Assume that supt∈½0;1k iðtÞk Land π : → = ð3Þ simple consequence. Let 2 G G G be the projection ho- that kξ′ðtÞ − ξ′ðtÞk 2 ; ≤ «. Then ð3Þ = ; ; 1 2 L ð½0 1Þ momorphism modulo G ½G ½G G, and let dπ2ðSÞ be the Stoll = ð3Þ π metric on G G associated to 2ðSÞ. ξ ; ξ ≤ «: deð 1ð1Þ 2ð1ÞÞ C Lemma 1. There is a constant C = CðSÞ > 0 such that for every ð3Þ Note that the only reason for splitting the original d∞-geodesic u ∈ G=G , there is γ ∈ Γ with ρ ðid; γÞ ≤ dπ ðid; uÞ and S 2ðSÞ into m pieces is to be able to have long enough pieces so that the dπ ðπ ðγÞ; uÞ ≤ C. 2ðSÞ 2 = ð3Þ ρ projection to G G can be well-approximated by a S-geodesic This last lemma will be helpful in the proof of Theorem 2 to using Lemma 1. Had we had Conjecture 5 at our disposal, both approximate a subFinsler geodesic by a word metric geodesic. parts of Proposition 6 would follow directly from Proposition 3, and in the second part, the coefficient αr would be as good as in 1 From Continuous Geodesics to Discrete Ones and Back Proposition 3, namely r. 1 Regarding Proposition 3 itself, we believe that the exponent r We now give a brief sketch of the proof of Theorem 2. For the 1 > sake of simplicity, we will assume that the Lie algebra g is iso- can be replaced with 2, even if r 2. We have checked this only when r ≤ 4 so far. Combined with Conjecture 5, having the ex- morphic to its graded g∞. Additional technicalities arise if this is = α = 1 not the case, but they do not affect our convergence rates. In ponent 1 2 would imply that one could take r 2 in Theorem 2 what follows we identify the Lie group with its Lie algebra via the also, at least for graded nilpotent groups. Stoll’s proof of Conjecture 5 for r = 2 relies on a good un- exponential map. On g∞ there is a natural one-parameter group δ fi derstanding of geodesics for the dS metric and in particular, the f tgt>0 of automorphisms called dilations and de ned by the formula δ ðxÞ = tix if x ∈ gðiÞ=gði+1Þ. The dilations scale the sub- fact that every point in G can be joined by a dS-geodesic that is t piecewise horizontal linear with a bounded number of distinct Finsler metric d∞ nicely and we have (Eq. 3) linear pieces. Despite the fact that Lemma 3 in (7) does not hold for r ≥ 3, this latter property is likely to hold for all r. d∞ðδtðxÞ; δtðyÞÞ = td∞ðx; yÞ: [3] Abnormal Geodesics, the Burago–Margulis Conjecture, and Theorem 2 can be easily deduced from the following propo- the Sharpness of Theorem 2 sition. We recall that BSðid; nÞ denotes the ball of radius n in Γ ρ We now pass to the second part of this note, which concerns with for the word metric S and d∞ is the associated Pansu limit fi Γ the construction of a speci c example showing that Theorem 2 is metric on the asymptotic cone of . sharp for groups of nilpotency class 2. The fact that the best α 1 exponent 2 is 2 instead of 1 is somewhat surprising, not only Proposition 6. Let n ∈ N, then, for αr as in Theorem 2, because α1 = 1 but because of the fact that, for the archetypal γ ∈ ; ∈ ; examples of step 2 groups, namely the Heisenberg groups, the (i) for every BSðid nÞ, there is y Bd∞ ðid 1Þ such that −1 best exponent is also 1 for any choice of generating set (15). d∞ y; δ1 ðγÞ = O n r , as n → ∞. n In fact this issue is related to another surprising phenomenon of subRiemannian geometry, namely the existence of abnormal ∈ ; γ ∈ ; (ii) for every x Bd∞ ðid 1Þ, there is x BSðid nÞ such that geodesics (see 5). Loosely speaking, these are geodesics, say −α ; δ γ = r → ∞ d∞ x 1 ð xÞ Oðn Þ, as n . connecting two points x and y, such that the end points of the n «-variations of that geodesic do not cover a full ball (for a Rie- mannian metric in some chart) of radius >C« around y, for some ≠ *Stoll’sdefinition of dS is slightly different, but it is easy to see that it yields the positive C. So typically, even when x y, if the geodesic connecting same distance. x and y is abnormal, one will be able to find points z near y at

Breuillard and Le Donne PNAS | November 26, 2013 | vol. 110 | no. 48 | 19223 Downloaded by guest on September 25, 2021 distance say « from y in a Riemannian metric that are not at Let d∞ be the Pansu limit metric on the asymptotic cone of ; + « Z ; ρ distance dðx yÞ Oð Þ from x. ðGð Þ iÞ. It is easy to see that Abnormal geodesics do not exist in Riemannian geometry as fi one can see from the rst variation formula, and they are a dis- d∞ðid; ðt; gÞÞ = jtj + d3ðid; gÞ; tinctive feature of subRiemannian geometry. A typical example is provided by segments of one-parameter horizontal subgroups in 1 where d3 is the subFinsler metric on H3ðRÞ associated to the ℓ the free nilpotent Lie group of step 2 and rank at least 3. norm jxj + jyj on the horizontal subspace RX⊕RY. See Fig. 1 for In this section, we consider the group G = R × H ðRÞ, the di- 3 a picture of the unit ball of ðH ðRÞ; d Þ. rect product of the 3-dimensional Heisenberg group and the 3 3 Theorem 7 is a simple consequence of the following proposi- additive group of R. We write an element of this group as tion, which shows the sharpness of the exponent 1 in Theorem 2 g = ðv; x; y; zÞ if g = ðv; expðxX + yY + zZÞÞ, where ðX; Y; ZÞ forms 2 for step-2 groups. a basis of the Lie algebra of H3ðRÞ such that ½X; Y = Z. It turns out that, if one puts a subFinsler product metric on G starting α = 1 1 = ρ ; ; 1 ρ R Proposition 8 (Sharpness of 2 2). Let Xn B 1 ðid nÞ 1 and with a subFinsler (but not Finsler) metric d3 on H3ð Þ, say := ; ; > n ; ; ; = + ; ; X∞ ðBd∞ ðid 1Þ d∞Þ. Then there is c 0 such that dðð0 idÞ ðt gÞÞ jtj d3ðg idÞ, then the segments fðt idÞgt∈½a;b are abnormal geodesics. Indeed, since Z is the central direction R ; « 1; > cffiffiffi: in the Lie algebrap offfiffiffi H3ð Þ, the points ð1 expð ZÞÞ are at dis- dGH Xn X∞ p tance at least 1 + c « from ð0; idÞ and not 1 + Oð«Þ as would be n the case had d3 been a Finsler (or Riemannian) metric on H3ðRÞ. 2 1 1 = ρ ; ; ρ For this reason, it is easy to cook up examples of subFinsler By contrast the convergence for Xn B 2 ðid nÞ n 2 is in O n , metrics d1 and d2 on G that are asymptotic and yet have jd1ðid; gÞ − hence, much faster. 1 The proof of Proposition 8 relies on some geometric consid- d2ðid; gÞj > Cd1ðid; gÞ2 for some C > 0 for arbitrarily large g,hence showing that the exponent 1 − 1 in Proposition 3 is sharp when r = 2. erations pertaining to the precise form of geodesics in the asymptotic r cone ðGðRÞ; d∞Þ and some of its finer geometry. The key to it, of ║ · ║ = + ; ; ║ · ║ = + Indeed, simply take 1 jvj maxfjxj jyj jzjg and 2 jvj course, is the existence of the abnormal geodesic t; 0; 0; 0 in ; fi fð Þgt∈½0;1 maxfjxj jyjg for the norms de ning the subFinsler metrics d1 and GðRÞ. In the next two sections, we give a sketch of the proof. d2. Note that, in this example, d1 is Finsler, while d2 is only sub- Finsler with horizontal subspace R × ðRX⊕RYÞ. Fine Geometry of the Heisenberg Group Equipped with the Z := An analogous example in the discrete group Gð Þ Pansu Metric Z × H3ðZÞ was given in (8) to disprove a conjecture of Burago ρ ρ We discuss here the geometry of the unit balls for the Pansu limit and Margulis. Let 1 and 2 be the left-invariant word metrics on R R × R GðZÞ induced by the finite generating sets metrics on H3ð Þ and on H3ð Þ and state the geometric ingredients needed in the proof of Proposition 8. n o Z ± ± ± ± The asymptotic cone of the discrete Heisenberg group H3ð Þ := ; ; ; 1; ; ; ; − 1; ; ; ; 1; ; ; ; 1 ± 1 ± 1 S1 ð1 0 0 1Þ ð1 0 0 1Þ ð0 1 0 0Þ ð0 0 1 0Þ endowed with standard generators fð1; 0; 0Þ ; ð0; 1; 0Þ g is the real Heisenberg group H3ðRÞ endowed with the subFinsler ℓ1 + and metric d3 induced by the norm jxj jyj on the horizontal sub- n o R ⊕R := ; ; ; ± 1; ; ; ; ± 1; ; ; ; ± 1 ; space X Y of the Lie algebra. A picture of its unit ball was S2 ð1 0 0 0Þ ð0 1 0 0Þ ð0 0 1 0Þ given in Fig. 1. This picture is borrowed from (8), where we R ; ρ ρ computed the precise form of geodesics in ðH3ð Þ d3Þ. respectively. Then, it follows from Theorem 4 that 1 and 2 are R ; ρ ;γ Geodesics in ðH3ð Þ d3Þ are horizontal paths and thus, can be 1ðid Þ γ + ∞ asymptotic in the sense that ρ ðid;γÞ tends to 1 as tends to described accurately by their projection to the ðx; yÞ plane [say Z ; ρ 2 Z ; ρ ; and thus that ðGð Þ 1Þ and ðGð Þ 2Þ have isometric asymp- ðxðtÞ yðtÞÞ]. There are three kinds of geodesics between id and γ = ; ; ; totic cones. However, if n ðn 0 0 nÞ, thenffiffiffi one checks easily a point g = ðx; y; zÞ ∈ H3ðRÞ. ρ ; γ = ρ ; γ − > p > that 1ðid nÞ n, while 2ðid nÞ n C n, for some C 0 (see Z ; ρ (i) geodesics of “staircase type,” where xðtÞ and yðtÞ are both 8). A picture of the unit ball of the asymptotic cone of ðGð Þ 1Þ is given in Fig. 2. monotone (see Fig. 4, curve c). This happens if and only if In (6) Burago and Margulis had conjectured that on any dis- jzj ≤ jxyj. Γ ρ ρ 2 crete group any two left-invariant word metrics 1 and 2 ρ ;γ (ii) 3-sided arcs of square with sides parallel to the x-axis and y- 1ðid Þ satisfying ρ ;γ → 1, as γ → ∞, must be at a bounded distance axis (see Fig. 4, curve a). This happens if and only if 2ðid Þ from each other, namely jρ ðid; γÞ − ρ ðid; γÞj ≤ C for all γ ∈ Γ. jxyj < ≤ ; 2 − jxyj 1 2 2 jzj maxfjxj jyjg 2 . Zd This is certainly the case in and Krat (15) established it for (iii) 4-sided arcs of the square with sides parallel to the x-axis and the Heisenberg group and for word hyperbolic groups. Abels y-axis (see Fig. 4 curve b). This happens if and only if and Margulis (12) proved an analogous result for word metrics > ; 2 − jxyj on reductive Lie groups. However, the above example shows that jzj maxfjxj jyjg 2 . it fails in general. This classification follows easily from the solution to Dido’s It turns out that a much stronger property holds for the word isoperimetric problem in the plane equipped with ℓ1 norm [see metrics ρ and ρ defined above on GðZÞ. We prove: 1 2 (16) and the Appendix to (8)]. The uniqueness issue for geodesics is easily addressed: geodesics of staircase type between id Theorem 7. Even though the two Cayley graphs are quasi-isometric = = and have isometric asymptotic cones, there is no C > 0 such that and g are never unique unless jzj jxyj 2. The 3-sided arcs of ðGðZÞ; ρ Þ is ð1; CÞ–quasi-isometric to ðGðZÞ; ρ Þ. square are unique and so are the 4-sided ones except if x or y is 0. 1 2 Accordingly, it is a simple matter to give an exact formula Recall that a map ϕ : X → Y between two metric spaces for d3.Weobtain: ðX; dX Þ and ðY; dY Þ is called a ð1; CÞ-quasi-isometry if jxyj (i)Ifjzj ≤ , then d3ðid; ðx; y; zÞÞ = jxj + jyj, dX ða; bÞ − C ≤ dY ðϕðaÞ; ϕðbÞÞ ≤ dX ða; bÞ + C, for all a; b ∈ X, and 2 every y ∈ Y is at distance at most C from some element of ϕðXÞ. jxyj ≤ ≤ ; 2 − jxyj ; ; ; = ; (ii)If 2 jzj maxfjxj jyjg 2 , then d3ðid ðx y zÞÞ maxfjxj This theorem is in sharp contrast with what happens in the jyjg + 2jzj ; Abelian case, where it is a simple matter to establish that two maxfjxj;jyjg qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zd ; 2 − jxyj ≤ ; ; ; = + jxyj − word metrics on have isometric asymptotic cones if and only if (iii)Ifmaxfjxj jyjg 2 jzj, then d3ðid ðx y zÞÞ 4 jzj 2 they are ð1; CÞ–quasi-isometric for some C > 0. jxj − jyj.

19224 | www.pnas.org/cgi/doi/10.1073/pnas.1203854109 Breuillard and Le Donne Downloaded by guest on September 25, 2021 pffiffiffi from genuine extreme points; in general they are only Oð «Þ away. The fact that this holds with Oð«Þ in the unit ball of SPECIAL FEATURE R × H3ðRÞ for the two points close to ð1; 0; 0; 0Þ and ð−1; 0; 0; 0Þ is a manifestation of the presence of the abnormal geodesic ; ; ; fðt 0 0 0Þgt∈½0;1 and is the heart of the matter here. The proof of Lemmas 3, 4, 5, and 6 relies on the above classification of geodesics in ðH3ðRÞ; d3Þ and the formulas for the distance d3 recalled above. Proof of Proposition 8 « := 1; fi – Set n dGH ðXn X∞Þ.Byde nition of the Gromov Hausdorff ; « ϕ : → 1 metric, there is a ð1 4 nÞ-quasi-isometry n X∞ Xn . Hence there are points x and y in X∞ that lie at a distance at most 4« n n n ; ; ; 1 ; ; ; − 1 from 1 0 0 n and 1 0 0 n , respectively. However, by Theo- ρ ; ; ; ; ; ; ; − ≅ ; ; ; ; ; ; ; − Fig. 4. Geodesics in the Pansu metric d on the Heisenberg group H ðRÞ.There rem 4, 1ððn 0 0 nÞ ðn 0 0 nÞÞ d∞ððn 0 0 nÞ ðn 0 0 nÞÞ 3 3 → + ∞ ; ; ; ; are three kinds of geodesics. The curve a is the projection of an example of (i.e., the ratio tends to 1) as npffiffiffi . However, d∞ððn 0 0 nÞ ; ; ; ; − = ; ; > > a geodesic with 3 sides. It connects ð0 0Þ to a point in the triangle with vertices ðn 0 0 nÞÞ dffiffiffi3ð0 0 2nÞ c n,forsomec 0. It follows that ; ; = ; ; = ; = p ð1 0Þ ð1 3 0Þ ð1 2 1 2Þ. The curve b is the projection of an example of d∞ðxn; ynÞ > c= n − 8«n. ; a geodesic with 4 sides. It connects ð0 0Þ to a point in the triangle with vertices Therefore, we are left to show that d∞ðxn; ynÞ = Oð«nÞ.Using ð0; 0Þ; ð1=3; 0Þ; ð1=2; 1=2Þ. The curve c is the projection of an example of a geo- Lemma 4, we claim that we may assume (up to precomposing each desic of staircase type. It connects ð0; 0Þ to a point ðx; 1 − xÞ with x ∈ ð0; 1Þ. ϕ → − ; ; ; n by the isometry v v) that xn and yn converge to ð1 0 0 0Þ. Indeed, the points ð1; 0; 0; 0Þ and ð−1; 0; 0; 0Þ are the only possible ϕ For more information on the geometry of polygonal sub- limit points of xn, since any subsequence of n has a subsequence fi Finsler metrics on the Heisenberg group, we refer the reader to converging to an isometry of X∞, which must either x the point ; ; ; fl − ; ; ; the nice recent preprint by Duchin and Mooney (17). ð1 0 0 0Þ or ip it to ð 1 0 0 0Þ according to Lemma 4. Let π denote the projection GðRÞ → R3 modulo the commutator The proof of Proposition 3 relies on a study of extreme points 3 subgroup. Note that πðX∞Þ is the ℓ1 unit ball in R and has 6 vertices in the unit balls of d3 and d∞. A collection of points g1; ...; gk in

; ; MATHEMATICS the unit ball is said to be a collection of extreme points if among which ð1 0 0Þ. Considering the 5 remaining vertices and the map π ∘ ϕ , we obtain points g ; ...; g ∈ X∞ such that dðgi; gjÞ = 2 for every i ≠ j. n 1 5 1 k ; ≥ − η ; ≥ − η η = « + 1 It is easy to see that in the unit ball for the ℓ norm in R , there d∞ðgi gjÞ 2 n and d∞ðgi xnÞ 2 n, where n 4 n n (ob- π ∘ ϕ η → ; ; ; is a unique collection of extreme points of size 2k, namely the serve that n is an n-submetry). Since xn ð1 0 0 0Þ as → + ∞ η vertices. We prove an analogous characterization of extreme n , it follows from Lemma 6 that xn is Oð nÞ close to ð1; 0; 0; 0Þ. The same applies to y obviously, and hence d∞ðx ; y Þ = points in H3ðRÞ and R × H3ðRÞ: n n n Oð«nÞ as desired. Lemma 3 (Extreme Points). Asymptotics for the Volume of Cayley Balls and Spheres ; ...; (i) Suppose g1 g4 is a collection of extremeÂ pointsÃ in the unit We now pass to the third part of this note and record some R ; ; ∈ 1; ball of ðH3ð Þ d3Þ.Thenthereareab 2 1 such that this applications of our main theorem to volume asymptotics for Cayley collection is fða; 1 − a; að1 − aÞÞ; ð1 − a; a; − að1 − aÞÞ; ð−b; balls and Cayley spheres in ﬁnitely generated nilpotent groups. − ð1 − bÞ; bð1 − bÞÞ; ð−ð1 − bÞ; − b; − bð1 − bÞÞg (see Fig. 5). In the asymptotic cone, the metric d∞ scales nicely under the (ii) Suppose g1; ...; g6 is a collection of extreme points in the unit dilation automorphisms, see (3). Hence the volume of balls R × R ; ; ; ; ; ; = d = ; ball of ð H3ð Þ d∞Þ. Then this collection is fð1 0 0 0Þ obeys the law volðBd∞ ðid tÞÞ Ct , where C volðBd∞ ðid 1ÞÞ. ; ð−1; 0; 0; 0Þ; ð0; g1Þ; ð0; g2Þ; ð0; g3Þ; ð0; g4Þg, where g1; ...; g4 Here d is the Hausdorff dimension of ðG∞ d∞Þ. It is an integer, are as in item (i). given by the Bass–Guivarc’h formula (18, 19) X This lemma has the following simple consequence for isome- d = kd ; = ; k tries of X∞ Bd∞ ðid 1Þ. k≥1 ϕ : → ; ; ; ; Lemma 4. Any isometry X∞ X∞ preserves the pair fð1 0 0 0Þ where d = dimgðkÞ=gðk+1Þ. Combined with Theorem 2, this gives: ð−1; 0; 0; 0Þg. k But the above classiﬁcation of extreme points is crucial to establish the following characterization of almost midpoints of extreme points.

Lemma 5 (Almost Midpoints of Extreme Points). Suppose we have 4 points h1; ...; h4 in Bd ðid; 1Þ such that d3ðhi; hjÞ ≥ 2 − « for every ≠ ∈ ; 3 ; + ; ≤ i j. Let p Bd3 ðid 1Þ be such that d3ðid pÞ d3ðid hiÞ d3ðp; hiÞ + « for every i = 1; ...; 4. Then d3ðid; pÞ = Oð«Þ as « → 0. Finally, Lemma 5 is used to establish the following main technical lemma needed in the proof of Proposition 8.

Lemma 6 (Almost Extreme Points). Suppose we are given 5 points ; ...; ; R = R × R ; ≥ g1 g5 in Bd∞ ðid 1Þ in Gð Þ H3ð Þ such that d∞ðgi gjÞ − « ≠ = ; ; ; ∈ ; 2 for all i j. Let g ðv x y zÞ Bd∞ ðid 1Þ be such that d∞ðg; giÞ ≥ 2 − « for every i = 1; ...; 5. Then as « → 0, either jvj = Oð«Þ, jv − 1j = Oð«Þ, or jv + 1j = Oð«Þ. ; Fig. 5. A collection of four extreme points in Bd3 ðid 1Þ. The extreme points R « ; ...; We note that, in the unit ball of H3ð Þ, -extreme points (i.e., h1 h4 are at distance 2 from one another and we show the projection of points h1; ...h4 as in Lemma 5) are not necessarily Oð«Þ away geodesics connecting them.

Breuillard and Le Donne PNAS | November 26, 2013 | vol. 110 | no. 48 | 19225 Downloaded by guest on September 25, 2021 Corollary 9 (Volume Asymptotics for Balls). Let Γ be a nilpotent group fi = −1 ∈ = n jSSðnÞj −K−d generated by a nite set S with S S and 1 S. Let BSðnÞ S be = O n ; ρ B n the ball of radius n centered at id for the word metric S induced by j Sð Þj Γ β > S. Let r be the nilpotency class of . Then, there is r 0 such that −d −β for any K > 4, which is not as good as our Corollary 11,4 < β = d + d r ; → ∞ r jBSðnÞj cSn OS n as n in general. β = ≤ β = 2 > and one can take r 1 if r 2 and r 3r if r 2. Concluding Remarks Γ Observe that Conjecture 5 reduces Conjecture 10 to the computation When is torsion-free, the constant cS above is the volume of of the asymptotics of the volume of large balls for the Stoll metric the unit ball of the asymptotic cone B ðid; 1Þ endowed with the d∞ d .Since,unliked∞, d does not satisfy the nice scaling property 3 Pansu limit metric, where the Haar measure is normalized so that S S π Γ π = π in general, it is not obvious that the asymptotics of the volume of the in the Abelianization, ð Þ has covolume one in ðGÞ ðG∞Þ. balls B ðid; tÞ has an error term of the form c td + Oðtd−1Þ. π : → = ; dS S Here, G G ½G G. However, one can prove that these balls, when scaled back by We recall that the asymptotics without error term was proved δ 1 ; the dilation 1, are only O away from Bd∞ ðid 1Þ in Hausdorff ≤ t t by Pansu in (2) and that the case r 2 is a result of Stoll (7). We distance (for a Riemannian metric). Thus, the volume asymp- believe that our error term for r > 2 is not sharp and that the ; totics for BdS ðid tÞ would follow from the following conjectural following holds: statement about the unit ball for the Pansu limit metric d∞ on ∞ d d−1 the asymptotic cone G . Conjecture 10. We have jBSðnÞj = cSn + OSðn Þ for all finitely generated nilpotent groups. Conjecture 12 (Regularity of subFinsler Spheres in Carnot Groups). The error term in the volume asymptotics for balls BSðnÞ in the The unit sphere of the Pansu metric d∞ is rectifiable with respect Γ Cayley graph of is, of course, related to the volume of spheres to any Riemannian distance on G∞. In particular, if the group G∞ = ∖ − SSðnÞ BSðnÞ BSðn 1Þ. Clearly, if one has the asymptotics has topological dimension n, the sphere has finite n − 1-dimensional = d + d−α α ≤ jBSðnÞj cSn Oðn Þ for some 1, then one also have Lebesgue measure. = d−α jSSðnÞj Oðn Þ. However, the knowledge of an upper bound In particular, the Pansu spheres ought not to be fractal sets. As it on the size of the spheres does not seem to give any information turns out, abnormal geodesics are also behind Conjecture 12 above; on the error terms in the volume of balls. in fact, they are the reason why this conjecture is not obvious, and hence, neither is the Oðtd−1Þ error term in the volume asymptotics Corollary 11 (Volume of Spheres). There are constants C1; C2 of t-balls for subFinsler metrics. Indeed, if there were no abnormal depending on S such that for all n ∈ N, we have geodesics, the distance function g ↦ d∞ðid; gÞ would be Lipschitz − −β and its level sets (the spheres) would be rectifiable. d 1 ≤ ≤ d r ; C1n jSSðnÞj C2n We recall incidentally that for certain Carnot–Carathéodory manifolds, the distance function and the spheres are known to be β where r is as in Theorem 9. not subanalytic (see 22). The upper bound follows immediately from Corollary 9,while Even if abnormal curves exist in most Carnot groups, they are the lower bound is a consequence of the following general fact: if conjectured to be sparse. According to Montgomery (ref. 5, chap. Γ fi ρ is any nitely generated group with word metric S, then 10.2), there ought to be a Sard theorem for the end-point map, im- plying in particular that the set of points in G∞ that can be reached by jBSðnÞj ≤ 2njSSðnÞj: a singular curve of length at most 1, say, must be a nowhere dense set of zero Lebesgue measure. This is still an open problem for general Corollary 11 improves on earlier results of Colding and Minicozzi Carnot groups. Should the answer be yes, it would then be possible to (ref. 20, lemma 3.3) (also rediscovered by Tessera in ref. 21) prove that subFinsler spheres are not fractal objects and that the bounding from above the volume of spheres in doubling metric n − 1-dimensional Lebesgue measure of subFinsler spheres is finite. spaces in terms of the doubling constant only. Pansu’s theorem d ACKNOWLEDGMENTS. E.B. thanks the European Research Council for its (i.e., jBSðnÞj ≅ cSn ) implies that nilpotent groups with word ρ support through Grant GADA-208091. E.B. and E.L.D. thank the Mathematical metric S are doubling metric spaces with doubling constant d Science Research Institute, Berkeley, for perfect working conditions during the ≤ð1 + «Þ2 for all balls of radius ≥rðS; «Þ, and in this case, their Quantitative Geometry program, when part of this research was conducted. argument gives an upper bound of the form: E.B. also thanks Fudan University, Shanghai, for its hospitality.

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