Products of involutions in the stable general linear group

TL;DR

This paper proves that in the stable general linear group over any field, every element with determinant ±1 can be expressed as a product of three involutions, with applications to automorphisms of infinite-dimensional vector spaces.

Contribution

It establishes the minimal number of involutions needed to factor elements with determinant ±1 in the stable general linear group over any field.

Findings

Every element with determinant ±1 is a product of three involutions.

No fewer than three involutions are sufficient for such factorizations.

Results apply to automorphisms of infinite-dimensional vector spaces as scalar multiples of finite-rank perturbations.

Abstract

In the stable general linear group over an arbitrary field, we prove that every element with determinant $\pm 1$ is the product of three involutions, and of no less in general. We also obtain several results of the same flavor, with applications to decompositions of automorphisms of an infinite-dimensional vector space that are scalar multiples of finite-rank perturbations of the identity.