Short words of infinite order

TL;DR

This paper establishes bounds on the shortest word length of elements of infinite order in finitely generated infinite linear groups, linking group properties to the Burnside problem and reflection groups.

Contribution

It provides a new upper bound for word length in infinite linear groups, with explicit calculations in degree two, advancing understanding of the Burnside problem.

Findings

Bound depends only on number of generators and degree

Explicit bound computed for degree two

Connects linear groups with reflection groups

Abstract

Given an infinite linear group with a finite set of generators, we show that the shortest word length of an element of infinite order has an upper bound that depends only on the number of generators and the degree. This provides a quantification of the Burnside problem for linear groups. In degree two, an explicit bound is computed using an exceptional connection to reflection groups.