Minimal dimension faithful linear representations of common finitely presented groups

TL;DR

This paper investigates the minimal dimension for faithful linear representations of certain finitely presented groups, revealing new bounds over complex numbers and fields of positive characteristic.

Contribution

It establishes new lower bounds for faithful linear representations of specific groups, including free by cyclic groups, over different fields.

Findings

Gersten's free by cyclic group has no faithful 4-dimensional representation over C

No faithful linear representation exists for this group over fields of positive characteristic

Bounds on minimal faithful representation dimensions are provided for various groups

Abstract

For various finitely presented groups, including right angled Artin groups and free by cyclic groups, we investigate what is the smallest dimension of a faithful linear representation. This is done both over C and over fields of positive characteristic. In particular we show that Gersten's free by cyclic group has no faithful linear representation of dimension 4 or less over C, but has no faithful linear representation of any dimension over fields of positive characteristic.