Super-expanders and warped cones

Sawicki, Damian

doi:10.5802/aif.3373

Cited by 7 publications

(2 citation statements)

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“…As such, we do not see how these methods could be relevant to Question 19, though it would be very interesting if they could be implemented for metric spaces that are not Banach spaces. Further examples of such expanders [163,39,55] were more recently obtained using Roe's warped cones [160] and a discretization procedure of Vigolo [178], but these take Lafforgue's expanders as input and therefore they suffer from the same deficiency vis-à-vis Question 19.…”

Section: Arbitrary Relative Expandersmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni, Naor and Neiman (2015). We establish this statement by proving that a metric space which is q-barycentric for some q ∈ [1,∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ ∞ grid [m] n ∞ = ({1,... ,m} n , · ∞ ) into ℓ q for all q ∈ (2,∞), from which we deduce the following discrete converse to the fact that ℓ n ∞ embeds with distortion O(1) into ℓ q for q = O(logn). A rigidity theorem of Ribe (1976) implies that for every n ∈ N there exists m ∈ N such that if [m] n ∞ embeds into ℓ q with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe's theorem does not give an explicit upper bound on this m, but by the work of Bourgain (1987) it suffices to take m = n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

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Section: Arbitrary Relative Expandersmentioning

confidence: 99%

Nonpositive curvature is not coarsely universal

Eskenazis

Mendel

2019

Invent. math.

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“…The first examples of super-expanders were constructed by Lafforgue in [Laf08] as a consequence of his work on strong Banach property (T) for SL 3 (F ), where F is a non-Archimedean local field. Since Lafforgue's work there have been several constructions of super-expanders using two main techniques: Namely, the work of Mendel and Naor on non-linear spectral calculus [MN14] which gave a zig-zag construction for super-expanders and, in a different direction, constructions using warped cones of groups actions arising from groups with strong Banach property (T) [Vig19,dLV19,FNvL19,Saw20].…”

Section: Introductionmentioning

confidence: 99%

Banach property (T) for $\rm SL_n (\mathbb{Z})$ and its applications

Oppenheim¹

2022

Preprint

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We prove a that for every n ≥ 3, the group SLn(Z) has a uniform version of Banach property (T) with respect to all super-reflexive Banach spaces. It readily follows that for every n ≥ 3, the group SLn(R) and all its lattices have a uniform version of Banach property (T) with respect to all super-reflexive Banach spaces.Consequences of this result are: First, for every n ≥ 4, the group SLn(R) and all their lattices have the Banach fixed point property with respect to all super-reflexive Banach spaces. Second, we settle a long standing open problem and show that the Margulis expanders (Cayley graphs of SLn(Z/mZ) for a fixed n ≥ 3) are super-expanders. Last, we deduce Banach property (T) and Banach fixed point property with respect to all superreflexive Banach spaces for a large family of higher rank algebraic groups.Our method of proof for Banach property (T) for SLn(Z) uses a novel result for relative Banach property (T) for the uni-triangular subgroup of SL3(Z) that is new even in the classical Hilbert setting and is interesting in its own right.

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Banach property (T) for $\mathrm{SL}_{n} (\mathbb{Z})$ and its applications

Oppenheim

2023

Invent. math.

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Super-expanders and warped cones

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org

Cited by 7 publications

References 30 publications

Nonpositive curvature is not coarsely universal

Nonpositive curvature is not coarsely universal

Banach property (T) for $\rm SL_n (\mathbb{Z})$ and its applications

Banach property (T) for $\mathrm{SL}_{n} (\mathbb{Z})$ and its applications

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