UNIFORM KAZHDAN CONSTANTS AND PARADOXES OF THE AFFINE PLANE
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Springer Science+Business Media New York (2020)
UNIFORM KAZHDAN CONSTANTS AND PARADOXES OF THE AFFINE PLANE LAM L. PHAM Department of Mathematics Yale University [email protected]
Abstract. Let G = SL(2, Z) n Z2 and H = SL(2, Z). We prove that the action G y R2 is uniformly non-amenable and that the quasi-regular representation of G on `2 (G/H) has a uniform spectral gap. Both results are a consequence of a uniform quantitative form of ping-pong for affine transformations, which we establish here.
1. Introduction 1.1. Kazhdan’s Property (T ) Let G be a countable group, and let S ⊂ G be a finite set. Given a unitary representation (π, H) of G, the Kazhdan constant (or spectral gap) of π relative to S is defined as κG (S, π) = inf sup kπ(g)ξ − ξk ξ ∈ Hπ , kξk = 1 . g∈S
If H ≤ G is a subgroup, we denote by HH the subspace of H-invariant vectors. We say that G has Kazhdan’s Property (T ) if there exists a finite set S generating G (henceforth, the group generated by S will be denoted by hSi) such that κG (S) = inf π κG (S, π) > 0, where the infimum is taken over all unitary representations (π, H) of G such that HG = {0}. An open problem first put forth by Lubotzky [28] is to determine for which groups inf S κG (S) > 0, where the infimum is taken over all finite sets S generating G. Such a group will be called uniform Kazhdan. When focusing on a specific representation π, let us write κG (π) = inf S κG (S, π). Gelander and Zuk [21] showed that a finitely generated group admitting a dense embedding in a connected Lie group cannot be uniform Kazhdan; this includes irreducible lattices in products of at least two Lie groups. On the other hand, Osin and Sonkin [36] managed to construct finitely generated uniform Kazhdan groups. While SL(3, Z) does have Property (T ), the problem of determining whether it is uniform Kazhdan remains open [4]. According to the Tits alternative [45], every finitely generated linear group is either virtually solvable, or contains a subgroup isomorphic to the non-abelian DOI: 10.1007/S00031-020-09600-5 Received April 29, 2019. Accepted January 23, 2020. Corresponding Author: Lam L. Pham, e-mail: [email protected]
LAM L. PHAM
free group on two generators F2 . Building on the work of Eskin, Mozes, and Oh [19], Breuillard and Gelander [11] showed that the Tits alternative could be made effective and uniform in the following sense: there exists N ∈ N such that for any finite symmetric generating set S (i.e., S −1 = S) containing 1, S N contains two generators of F2 (Eskin, Mozes, and Oh [19] proved uniform exponential growth by finding generators of a free subsemigroup). Recall that a discrete group G is uniformly non-amenable if κG (λG ) > 0 where λG is the left regular representation of G on `2 (G); this was first investigated by Shalom [41] and Osin [37], and a slightly different definition was given in [2]. One of the key applications of the uniform Tits alternative is precisely to show that non-virtually solvable finitely generated linear groups are uniformly non-amenab
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