Roundness in Analysis and Topology Stratos Prassidis Canisius College

Roundness in Analysis and Topology – p. 1 Motivations and Definitions

Theorem. Every Banach space is homeomorphic to a Hilbert space.

Problem. Let X and Y be Banach space that are uniformly homeomorphic. Are they linearly isomorphic?

Ribe: No, they are not even Lipschitz isomorphic.

Johnson, Linderstauss, Schectman: Yes, if Y = ℓp, 1 < p < ∞. Unknown for large class of Banach spaces such as Lp[0, 1], ℓ1, ℓ∞.

Roundness in Analysis and Topology – p. 2 Problem. When are two Banach spaces uniformly homeomorphic?

Linderstauss (1963), Enflo (1968): If p, q ≥ 1 with p =6 q, then Lp(µ) is not uniformly homeomorphic to Lq(ν).

Except in the case p = q′ the theorem uses roundness.

Roundness in Analysis and Topology – p. 3 Definition (Enflo): A (quasi)metric space (X,d) is said to have roundness p if p is the supremum of all q that satisfy: For all {x00,x10,x01,x11}

q q d(x00,x11) + d(x01,x10) ≤ q q q q d(x00,x10) + d(x00,x01) + d(x01,x11) + d(x10,x11) .

Roundness in Analysis and Topology – p. 4 Per Enflo: The inequality holds for n-cubes. More n precisely, let {xε} be a collection of 2 points such that ε = (ε1,...,εn) where εi ∈{0, 1}. A pair (xε,xδ) is called an edge if the indices differ in exactly one coordinate and it is called a diagonal if the indices differ in all coordinates. Then X has roundness p iff the sum of the p-th powers of the diagonals is less than or equal to the sum of the p-th powers of the sides. Notice that there are 2n−1 diagonals and n2n−1 edges.

Roundness in Analysis and Topology – p. 5 Remarks: If (X,d) is a metric space, then p ≥ 1, from the triangle inequality. If there is a pair of points of X that have a midpoint, then p ≤ 2. If the Banach space satisfies the parallelogram law, then p =2. If p < ∞, the inequality holds for q = p.

Roundness in Analysis and Topology – p. 6 Results

Enflo: round(Lp(µ)) = p, for 1 ≤ p ≤ 2. ′ Weston: round(Lp(µ)) = p , for 2 ≤ p ≤ ∞.

Weston: round(Lp(µ)) = p, for 0

Roundness in Analysis and Topology – p. 7 Back to the Uniform Homeomorphism Problem: Lp[0, 1] and ℓp are not uniformly homeomorphic: Enflo (1970): p =1. Bourgain (1984): 1 < p < 2. Gorelik (1994): 2 < p < ∞. Weston (1993): 0 < p < 1.

Roundness in Analysis and Topology – p. 8 Tools used to prove such theorems:

(1) Corson-Klee Lemma. Let f : M → Y be uniformly continuous map with M a convex subset of a Banach space and Y a metric space. The f is Lipschitz of order 1 for large distances:

∀δ > 0, ∃K(δ),d(f(x),f(y)) ≤ K(δ)kx − yk, for kx − yk≥ δ.

(2) Linderstrauss, Enflo: Let f : Lp(µ) → X be a uniform homeomorphism with 1 ≤ p ≤ 2 and X a metric space of roundness q>p then f cannot satisfy a Lipschitz condition of order < (q/p) for large distances.

Roundness in Analysis and Topology – p. 9 More Results

Definition: A uniform Banach Group G is a Banach space X equipped with a group operation that is uniformly continuous and the zero element of X is the identity element of the group (Example: (X, +)).

Problem: Are there non-commutative uniform Banach groups?

Roundness in Analysis and Topology – p. 10 Remarks If φ : X → F be a uniform homeomorphism from a Banach space to a locally bounded linear space with φ(0) = 0, then

x·y = φ−1(φ(x) + φ(y))

defines a uniform Banach group structure on X. For a uniform Banach group structure G on X,

d(x, y) = sup kwxz − wyzk w,z∈G

defines a G-invariant metric on X.

Roundness in Analysis and Topology – p. 11 The intrinsic distance on X is defined as

n−1 dI (x, y) = inf d(xi,xi+1) Xi=1 n where {xi}i=1 is a chain of points with x0 = x, xn = y and d(xi,xi+1) ≤ 1. The intrinsic distance is uniformly equivalent to the norm.

Roundness in Analysis and Topology – p. 12 P. - Weson: Let φ : X → F be a uniform homeomorphism between a Banach space of non-trivial roundness and a quasi-normed space and G the uniform Banach group structure induced on X. Define a Banach space structure on G by: zt = φ−1(φ(z)), d (zt, 0) N(z) = lim sup I . t→∞ t Then (G, N) is uniformly homeomorphic to X and linearly isomorphic to F . In particular, F is normable with norm M(y) = N(φ−1(y)).

Roundness in Analysis and Topology – p. 13 Main Points of the Proof. Since the roundness of X is p > 1, there is K > 0 such that for each z ∈ G, dI (z, 0) ≥ 1, there is u ∈ G satisfying:

2 2 1/p dI (u , z) ≤ K, |2dI (u, 0) − dI (u , 0)|≤ KdI (z , 0).

Using the previous result, if di(v, 0) ≥ 1,

2n n n 1/p |dI (v , 0) − 2 dI (v, 0)|≤ 2 KdI (v, 0) Then

2n dI (v , 0) 1/p dI (v, 0) N(v) ≥ lim sup n ≥ dI (v, 0)−KdI (v, 0) ≥ . n→∞ 2 2 Roundness in Analysis and Topology – p. 14 Then Consequences

(Orlicz spaces) LΦ is not uniformly homeomorphic to LΨ if there is K and p > 1 such that for s ≥ 0 and λ ≥ 1, Ψ(t) Φ(λs) ≥ K·λp′·Φ(s), lim =0. t→∞ t

Lp(µ) and Lq(ν) are not uniformly homeomorphic if 0 ≤ q

1 ≤ min(p, q) ≤ 2, or 0 ≤ q < 1 < p.

Roundness in Analysis and Topology – p. 15 Definitions. A Banach space E has type p if there is a constant C such that:

p 1/p 1/p 1 n n p ri(t)xi dt ≤ C kxik . 0 ! ! Z i=1 i=1 X X

(1 ≤ p ≤ 2) . A Banach space E has cotype q if there is a constant C such that (q ≥ 2):

q 1/q 1/q 1 n n q ri(t)xi dt ≥ C kxik . 0 i=1 ! i=1 ! Z Roundness in Analysis and Topology – p. 16 X X

Remarks: The Rademacher functions on [0, 1] are defined as:

j rj(t) = sign(sin(2 πt).

(Khintchine’s Inequality) Given 0 < p < ∞, there are constants Ap and Bp such that

p 1/p 2 1/2 1 n 1 n A a r (t) dt ≤ a r (t) dt p  j j   j j  Z0 j=1 Z0 j=1 X  X   p  1/p 1 n  

≤ Bp ajri(t) dt 0 i=1 ! Z Roundness in Analysis and Topology – p. 17 X

A Banach space E is a Hilbert space iff it has type and cotype equal to 2. (Bourgain-Milman-Wolfson) A metric space (X,d) has metric type p if there is a constant B such that for all n-cubes in X:

1/2 1/2 1 − 1 l(d)2 ≤Bn p 2 l(e)2 .     d,diagonalX e,Xedge     (Bourgain-Milman-Wolfson) A metric space has non-trivial metric type iff it does not contain uniformly Lipschitz images of n-dimensional Hamming cubes.

Roundness in Analysis and Topology – p. 18 A Banach space has type > 1 iff it has metric type > 1. Thus if a Banach space contains finite dimensional n Hamming cubes uniformly, then it contains ℓ1 ’s uniformly. For Banach space, roundness p implies metric type p (2) with B =1. Furthermore, ℓ1 ⊕∞ℓ2 has roundness 1 and type greater than 1. The main property used in the proof of the main theorem is that, for any n cube,

1/p l(dmin) ≤ B·n ·l(emax).

Roundness in Analysis and Topology – p. 19 M. Mendel and A. Naor gave the appropriate definition of metric cotype and proved its properties. One sample result: If a Banach space Y coarsely embed into a Banach space X of non-trivial cotype, then qY ≤ qX .

Roundness in Analysis and Topology – p. 20 Geometric Results

Lafont - P. The roundness of a circle with the arc length is 1. If X is a complete metric space and the infimum of the lengths of all the homotopically non-trivial curves is realized, then X has roundness 1. A compact non-simply connected Riemannian manifold has roundness 1.

Roundness in Analysis and Topology – p. 21 CAT(0)-spaces have roundness 2 (Uses comparison theorems for quadrilaterals). If the roundness of a geodesic metric space X is 2, then there is a unique geodesic joining them. Furthermore, if X is proper then X is contractible. Locally finite trees have roundness 2. Roundness is not a quasi-isometry invariant.

Roundness in Analysis and Topology – p. 22 Definition For a group G, define the roundness spectrum

ρ(G) = {round(ρ(Cay(G, Σ)) : Σ generates G}⊂ [1, ∞]

Remarks If G is infinite finitely generated group then ρ(G) ⊂ [1, 2]. If G is finite ∞∈ ρ(G).

Roundness in Analysis and Topology – p. 23 Let G be a group that contains two elements x and y such that:

x2 =16 =6 y2, x =6 y±1 =6 x3, y3 =6 x±1.

Then 1 ∈ ρ(G). Proof. Include x and y in a generating set Σ of G. Remove x2 and y2 if they belong to Σ. Include generators:

−1 −1 −1 −1 z1 = x y, z2 = xy, z3 = xy , z4 = x y .

Then the quadrilateral [x,y,x−1, y−1] has all edges of length 1 and the diagonals [x,x−1], [y, y−1] have length 2. That configuration has roundness 1.

Roundness in Analysis and Topology – p. 24 If G is finitely generated group with 1 ∈/ ρ(G). Then G is a torsion group with torsion of order 2, 3, 5 or 7. Proof. If G has torsion bigger than 7 and g ∈ G. Choose g′ ∈ hgi such that:

g′ ∈{/ g,gn−1,g3,gn−3}, (g′)3 =6 g±1 and use the last statement with g and g′. If g has order 4, then include g in the generating set and remove g2 and g4. Then [1,g,g2,g4] has roundness 1. If g has order 6, then g and g2 satisfy the conditions of the last statement. ρ(Zn) = {1} for n ≥ 2.

Roundness in Analysis and Topology – p. 25 Open Speculations/Questions. If G is δ-hyperbolic then ρ(G) is dense in [1, 2]. If G contains Z2 as a quasi-convex subgroup then ρ(G) = {1}.

Roundness in Analysis and Topology – p. 26 Genralized Roundness

Definition. A metric space (X,d) has generalized roundness p if p is the supremum of all q such that:

q q q (d(ai, aj) + d(bi,bj) ) ≤ d(ai,bj) ≤ ≤ ≤ ≤ 1 Xi

Roundness in Analysis and Topology – p. 27 Remarks. For any (quasi)metric space X, its generalized roundness is ≥ 0. Roundness is obtained from generalized roundness from double 2-simplices. (Enflo) For all 1 ≤ p ≤ 2, the generalized roundness of Lp(µ) is p.

(Lennard-Tonge-Weston) If Lp(µ), 2

Roundness in Analysis and Topology – p. 28 Smirnov’s Problem. Is every separable metric space uniformly homeomorphic to a subset of L2[0, 1].

Enflo’s Counterexample: A metric space M such that every separable metric space is uniformly homeomorphic to a subset of M has generalized roundness 0. But L2[0, 1], being a Hilbert space, has generalized roundness 2.

Roundness in Analysis and Topology – p. 29 Geometric Applications

The connection between generalized roundness and geometry is give by the following:

Lennard-Tonge-Weston: A metric space (X,d) has generalized roundness p > 0 iff the function dp is a negative definite kernel: n p d (xi,xj)aiaj ≤ 0, when aj =0. ≤ ≤ 1 Xi,j n Xj=1

Roundness in Analysis and Topology – p. 30 Lafont-P. Let Γ be a discrete Kazhdan infinite group and Σ a finite generating set. The the generalized roundness of Cay(Γ, Σ) is 0. It follows from the work of P. Delorme, A. Guichardet, P. de la Harpe and A. Valette. If Γ is a finitely generated group that is coarsely embedded into a metric space X of positive generalized roundness. Then Γ coarsely embeds into a Hilbert space and thus it satisfies the coarse Baum–Connes Conjecture and thus the Novikov Conjecture. Proof.(X coarsely embeds into H) =⇒ (Γ coarsely embed into H). Done by Yu’s results. Roundness in Analysis and Topology – p. 31 Consequences 1. Let Γ be a discrete group and X a metric space with positive generalized roundness, such that either Γ embeds isometrically on X, or Γ acts properly discontinuously, by isometrics, with finite stabilizers on X Then Γ satisfies the coarse Baum–Connes Conjecture.

2. If Γ embeds isometrically into Lp(µ) with 1 ≤ p ≤ 2, then Γ satisfies the coarse Baum–Connes Conjecture. Similar results are were proved by P. Nowak, W. B. Johnson and N. L. Randrianarinovy. 3. There are Kazhdan groups (like uniform lattices in Sp(n, 1)) that are coarsely embeddable into a Hilbert space and they do not satisfy any of the conditions above. Roundness in Analysis and Topology – p. 32 More Questions

(1) Is every CAT(0) space coarsely equivalent to a space of positive generalized roundness?

J. Faraut and K. Harzallah proved that the quartenionic hyperbolic space has trivial generalized roundness. An affirmative answer to the question will imply that all the groups that act geometrically on CAT(0)-spaces satisfy the coarse Baum–Connes Conjecture (they already satisfy the Novikov Conjecture by Carlsson–Pedersen and Farrell–Lafont).

Roundness in Analysis and Topology – p. 33 (2) Does every compact Riemannian manifold contain a globally minimizing closed geodesic?

That will imply that all compact Riemannian manifolds have roundness 1.

Roundness in Analysis and Topology – p. 34 (3) For Cayley graphs: What is the generalized roundness of Zn? Characterize the groups which admit a presentation with Cayley graph of positive generalized curvature. Let X and Y be two discrete metric spaces with bounded geometry and distances bounded away from zero, that are bi-Lipschitz equivalent. Then the generalized roundness of X is positive iff the generalized roundness of Y is positive.

Roundness in Analysis and Topology – p. 35