## Abstract

An invariant random subgroup of the countable group Γ is a random subgroup of Γ whose distribution is invariant under conjugation by all elements of Γ. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on Γ is strictly less than the spectral radius of the corresponding random walk on Γ=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.

Original language | English |
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Pages (from-to) | 465-488 |

Number of pages | 24 |

Journal | Duke Mathematical Journal |

Volume | 163 |

Issue number | 3 |

DOIs | |

State | Published - 15 Feb 2014 |

## ASJC Scopus subject areas

- Mathematics (all)