Infinite-dimensional general linear groups are groups of universally finite width

TL;DR

This paper confirms that infinite-dimensional linear groups over division rings have the property of universally finite width, meaning any generating set can produce the entire group with a bounded number of elements.

Contribution

It proves Bergman's conjecture that infinite-dimensional linear groups over division rings possess universally finite width, extending the property from symmetric groups to these linear groups.

Findings

Infinite-dimensional linear groups over division rings have universally finite width.

Any generating set of such a group can generate the entire group with a bounded number of elements.

Supports Bergman's conjecture for a new class of infinite groups.

Abstract

Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set $\Omega,$ there is a uniform bound $k \in \N$ such that any permutation $\sigma \in \text{Sym}(\Omega)$ is a product of at most $k$ elements of $X \cup X^{-1},$ or, in other words, $\text{Sym}(\Omega)=(X^{\pm 1})^k.$ Bergman also formulated a sort of general conjecture stating that `the automorphism groups of structures that can be put together out of many isomorphic copies of themselves' might be groups of universally finite width and particularly mentioned, in this respect, infinite-dimensional linear groups. In this note we confirm Bergman's conjecture for infinite-dimensional linear groups over division rings.