Words have bounded width in $SL(n,\mathbb{Z})$

TL;DR

This paper proves that in the special linear group over integers, the width of any word is uniformly bounded for large enough dimensions, with explicit bounds depending on the dimension.

Contribution

It establishes uniform bounds on the word width in $SL_n(Z)$, showing that the width is bounded independently of the specific word for sufficiently large n.

Findings

For every n ≥ 3, there exists a constant C(n) bounding the width of any word in $SL_n(Z)$.

The width of any word in $SL_n(Z)$ is at most 87 when n is sufficiently large.

The results provide explicit bounds on word width in high-dimensional special linear groups.

Abstract

We prove two results about width of words in $SL_n(\mathbb{Z})$. The first is that, for every $n \geq 3$, there is a constant $C(n)$ such that the width of any word in $SL_n(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $SL_n(\mathbb{Z})$ is at most 87.