## Duke Mathematical Journal

### Kesten’s theorem for invariant random subgroups

#### Abstract

An invariant random subgroup of the countable group $\Gamma$ is a random subgroup of $\Gamma$ whose distribution is invariant under conjugation by all elements of $\Gamma$. We prove that for a nonamenable invariant random subgroup $H$, the spectral radius of every finitely supported random walk on $\Gamma$ is strictly less than the spectral radius of the corresponding random walk on $\Gamma/H$. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that, for a Cayley graph $G$ of a linear group with no amenable normal subgroups, any sequence of finite quotients of $G$ that spectrally approximates $G$ converges to $G$ in Benjamini–Schramm convergence. In particular, this implies that infinite sequences of finite $d$-regular Ramanujan–Schreier graphs have essentially large girth.

#### Article information

Source
Duke Math. J., Volume 163, Number 3 (2014), 465-488.

Dates
First available in Project Euclid: 11 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1392128875

Digital Object Identifier
doi:10.1215/00127094-2410064

Mathematical Reviews number (MathSciNet)
MR3165420

Zentralblatt MATH identifier
1344.20061

#### Citation

Abért, Miklós; Glasner, Yair; Virág, Bálint. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163 (2014), no. 3, 465--488. doi:10.1215/00127094-2410064. https://projecteuclid.org/euclid.dmj/1392128875

#### References

• [1] M. Abért, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, and I. Samet, On the growth of Betti numbers of locally symmetric spaces, C. R. Math. Acad. Sci. Paris 349 (2011), 831–835.
• [2] M. Abért, Y. Glasner, and B. Virág, The measurable Kesten theorem, preprint, arXiv:1111.2080v2 [math.PR].
• [3] D. Aldous and R. Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (2007), 1454–1508.
• [4] I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), 13 pp.
• [5] N. Bergeron and D. Gaboriau, Asymptotique des nombres de Betti, invariants $l^{2}$ et laminations, Comment. Math. Helv. 79 (2004), 362–395.
• [6] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
• [7] C. Chabauty, Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. France 78 (1950), 143–151.
• [8] J. Friedman, “A proof of Alon’s second eigenvalue conjecture” in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (San Diego, 2003), ACM, New York, 2003, 720–724.
• [9] Y. Glasner, Invariant random subgroups of linear groups, in preparation.
• [10] F. P. Greenleaf, Invariant Means on Topological Groups and their Applications, Van Nostrand Mathematical Studies 16, Van Nostrand Reinhold, New York, 1969.
• [11] R. I. Grigorchuk, “Symmetrical random walks on discrete groups” in Multicomponent Random Systems, Adv. Probab. Related Topics 6, Dekker, New York, 1980, 285–325.
• [12] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156.
• [13] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354.
• [14] A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.
• [15] G. A. Margulis, Finiteness of quotient groups of discrete subgroups (in Russian), Funktsional. Anal. i Prilozhen. 13, no. 3 (1979), 28–39; English translation in Funct. Anal. Appl. 13, no. 3 (1979), 178–187.
• [16] M. Morgenstern, Existence and explicit constructions of $q+1$ regular Ramanujan graphs for every prime power $q$, J. Combin. Theory Ser. B 62 (1994), 44–62.
• [17] A. Nilli, On the second eigenvalue of a graph, Discrete Math. 91 (1991), 207–210.
• [18] R. Ortner and W. Woess, Non-backtracking random walks and cogrowth of graphs, Canad. J. Math. 59 (2007), 828–844.
• [19] W. L. Paschke, Lower bound for the norm of a vertex-transitive graph, Math. Z. 213 (1993), 225–239.
• [20] G. Stuck and R. J. Zimmer, Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2) 139 (1994), 723–747.
• [21] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.
• [22] A. M. Vershik, Totally nonfree actions and the infinite symmetric group, Mosc. Math. J. 12 (2012), 193–212.
• [23] B. A. F. Wehrfritz, Infinite Linear Groups: An Account of the Group-theoretic Properties of Infinite Groups of Matrices, Ergeb. Math. Grenzgeb. (3) 76, Springer, New York, 1973. xiv+229 pp.