Linear representations of random groups

TL;DR

This paper proves that Gromov random groups at any positive density lack non-trivial degree-k linear representations over any field, contrasting with known results for lower densities where faithful linear representations exist.

Contribution

It establishes a new non-existence result for linear representations of random groups at positive densities, extending understanding of their algebraic properties.

Findings

Random groups at positive density have no non-trivial degree-k representations over any field.

Contrasts with low-density cases where faithful linear representations over rationals exist.

Highlights a phase transition in linear representability based on density.

Abstract

We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the density is less than 1/6 such groups have a faithful linear representation over the rationals, a.a.s.