On the spectral gap for finitely-generated subgroups of \(\text{SU}(2)\)
type: Item
instance of: scholarly article
title: On the spectral gap for finitely-generated subgroups of \(\text{SU}(2)\)
zbMATH Open document ID: 1135.22010
published in: Inventiones Mathematicae
publication date: 2008-02-18T00:00:00Z
review text: Let \(g_1,\dots ,g_k\), \(k\geq 2\), be elements of the group \(G=\text{SU}(2)\). Consider the averaging operator \[ z=z_{g_1,\dots ,g_k}=\sum_{j=1}^k (f(g_jx)+f(g_j^{-1}x)). \] This operator is selfadjoint on \(L_2(G)\), the constant function is its eigenfunction with the eigenvalue \(\lambda_0=2k\). Denote by \(\lambda_1\) the supremum of the eigenvalues of \(z\) on the orthogonal complement of the constant functions. The element \(z\) is said to have a spectral gap if \(\lambda_1<2k\). The authors prove the spectral gap property for elements satisfying a noncommutative diophantine condition. In particular, if \(g_1,\dots ,g_k\) generate a subgroup containing elements with algebraic entries, which generate a non-Abelian free group, then \(z\) has a spectral gap. The authors describe connections of the problem and their techniques with the problem of uniqueness of the finitely additive rotation invariant measure, the Katz-Tao discretized ring conjecture (proved by Bourgain), etc. Open problems are formulated.
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A75
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 5236752
zbMATH Keywords: spectral gap property
zbMATH Keywords: noncommutative diophantine condition
author: Jean Bourgain
author: Alex Gamburd
MaRDI profile type: Publication
full work available at URL: s00222-007-0072-z
OpenAlex ID: W2031808668
cites work: On the Erdős-Volkmann and Katz-Tao ring conjectures
cites work: Uniform expansion bounds for Cayley graphs of \(\text{SL}_2(\mathbb F_p)\).
cites work: A sum-product estimate in finite fields, and applications
cites work: Quaquaversal tilings and rotations
cites work: Q3522568
cites work: Free groups in linear groups
cites work: Finitely additive measures on \(S^ 2\) and \(S^ 3\), invariant with respect to rotations
cites work: Growth rates in the quaquaversal tiling
cites work: Almost all subgroups of a Lie group are free
cites work: Spectra of elements in the group ring of SU(2)
cites work: Q3856819
cites work: Efficient discrete approximations of quantum gates
cites work: Growth and generation in \(\text{SL}_2(\mathbb{Z}/p\mathbb{Z})\).
cites work: Diophantine properties of elements of \(SO(3)\)
cites work: Some connections between Falconer's distance set conjecture and sets of Furstenburg type
cites work: Symmetric Random Walks on Groups
cites work: Hecke operators and distributing points on the sphere I
cites work: Hecke operators and distributing points on S2. II
cites work: Subgroups of \(\text{SO}(3)\) associated with tilings
cites work: On 2-Generator Subgroups of SO(3)
cites work: Q3999365
cites work: Bounds for multiplicities of automorphic representations
cites work: For 𝑛>3 there is only one finitely additive rotationally invariant measure on the 𝑛-sphere defined on all Lebesgue measurable subsets
cites work: A class of free rotation groups
cites work: Q5393666
DOI: 10.1007/S00222-007-0072-Z
description: scientific article