Lattices with and lattices without spectral gap

TL;DR

This paper investigates the spectral gap properties of lattices in simple algebraic groups over local fields, showing conditions under which spectral gaps exist or do not exist, thus answering a question posed by Margulis.

Contribution

It proves that lattices in certain algebraic groups have spectral gaps, but also constructs examples where no spectral gap exists, resolving a long-standing open question.

Findings

Lattices in simple algebraic groups over local fields have spectral gaps.

Existence of lattices in automorphism groups of regular trees without spectral gaps.

Answers negatively to Margulis's question about spectral gaps in lattices.

Abstract

The following two results are shown. 1) Let $G$ be the $k$-rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G.$ Then the regular representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean). 2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a $k$-regular tree for $k>2.$