Characterizing linear groups in terms of growth properties

TL;DR

This paper explores how growth functions related to residual finiteness can be used to characterize linear groups, including hyperbolic groups, by analyzing their approximation properties via finite quotients.

Contribution

It introduces new growth functions that serve as criteria for linearity in groups, extending the understanding of residual finiteness in hyperbolic and related groups.

Findings

Growth functions characterize linearity in certain groups

Residual finiteness growth relates to group linearity

Hyperbolic groups are included in the characterization

Abstract

Residual finiteness growth measures how well-approximated a group is by its finite quotients. We prove that some related growth functions characterize linearity for a class of groups including all hyperbolic groups.