Strongly convergent unitary representations of right-angled Artin groups

TL;DR

This paper introduces a novel random matrix model demonstrating that right-angled Artin groups and related groups have finite-dimensional unitary representations that strongly converge to their regular representations, impacting geometric and spectral properties.

Contribution

It presents a new random matrix approach to establish strong convergence of unitary representations for right-angled Artin groups and related classes.

Findings

All right-angled Artin groups have finite-dimensional unitary representations converging to the regular representation.

The result extends to fundamental groups of certain hyperbolic manifolds, Coxeter groups, and hyperbolic cubulated groups.

Implication: sequences of flat Hermitian vector bundles on hyperbolic 3-manifolds have Laplacian spectra approaching at least 1.

Abstract

We prove using a novel random matrix model that all right-angled Artin groups have a sequence of finite dimensional unitary representations that strongly converge to the regular representation. We deduce that this result applies also to: the fundamental group of a closed hyperbolic manifold that is either three dimensional or standard arithmetic type, any Coxeter group, and any word-hyperbolic cubulated group. One strong consequence of these results is that any closed hyperbolic three-manifold has a sequence of finite dimensional flat Hermitian vector bundles with bottom of the spectrum of the Laplacian asymptotically at least 1.