Coarse non-amenability and covers with small eigenvalues

Arzhantseva, Goulnara; Guentner, Erik

doi:10.1007/s00208-011-0759-8

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…However even in the large injectivity radius case, h(X n ) (and hence the spectral gap) need not be bounded from below. It is possible to construct towers of coverings X n such that λ (n) 1 → 0 (See for example [15,7]). 1.3.…”

Section: 2mentioning

confidence: 99%

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Masson¹,

Sahlsten²

2017

Duke Math. J.

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Abstract. We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

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Section: 2mentioning

confidence: 99%

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Masson¹,

Sahlsten²

2017

Duke Math. J.

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“…Here we show that H γ may weakly contain the trivial representation for a certain sequence of finite index subgroups and for certain intermediate γ. We follow here [1].…”

Section: 2mentioning

confidence: 99%

“…n , be the iterated subgroups generated by the squares of all elements of the previous group. There is a nice description of X n = F 2 /G n in [1] as vertices of Cayley graphs of X n . The Cayley graph of F 2 /G 0 is the wedge of two circles with a single vertex, and the Cayley graphs Cay(X n ) of F 2 /G n = X n are constructed from it inductively.…”

Section: 2mentioning

confidence: 99%

On a Family of Representations of Residually Finite Groups

Manuilov

2019

Algebr Represent Theor

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For a residually finite group G, its normal subgroups G ⊃ G 1 ⊃ G 2 · · · with ∩ n∈N G n = {e} and for a growth function γ we construct a unitary representation π γ of G. For the minimal growth, π γ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients G/G n . In the case of intermediate growth we show two examples of different behaviour of π γ .

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“…Using part (ii) of Corollary 1.2 we prove that there is no coarse embedding of the box spaces of SL n (Z) into the box spaces of SL m (Z), where n > m and n, m ≥ 3. Some considerations of non coarsely equivalent box spaces can be found in [2].…”

Section: Applicationsmentioning

confidence: 99%

From the geometry of box spaces to the geometry and measured couplings of groups

Das

2018

J. Topol. Anal.

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In this paper, we prove that if two 'box spaces' of two residually finite groups are coarsely equivalent, then the two groups are 'uniform measured equivalent' (UME). More generally, we prove that if there is a coarse embedding of one box space into another box space, then there exists a 'uniform measured equivalent embedding' (UME-embedding) of the first group into the second one. This is a reinforcement of the easier fact that a coarse equivalence (resp. a coarse embedding) between the box spaces gives rise to a coarse equivalence (resp. a coarse embedding) between the groups.We deduce new invariants that distinguish box spaces up to coarse embedding and coarse equivalence. In particular, we obtain that the expanders coming from SL n (Z) can not be coarsely embedded inside the expanders of SL m (Z), where n > m and n, m ≥ 3. Moreover, we obtain a countable class of residually groups which are mutually coarse-equivalent but any of their box spaces are not coarseequivalent.

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Coarse non-amenability and covers with small eigenvalues

Cited by 4 publications

References 12 publications

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

On a Family of Representations of Residually Finite Groups

From the geometry of box spaces to the geometry and measured couplings of groups

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