Several remarks on groups of automorphisms of free groups

TL;DR

This paper explores the structure of automorphism groups of infinite free groups, showing that certain double cosets form a semigroup and analyzing their actions on product spaces of compact groups.

Contribution

It introduces a natural semigroup structure on double cosets of automorphism groups and studies their actions on $L^2$ spaces of product groups.

Findings

Double cosets form a natural semigroup structure.

Automorphism groups act on $L^2$ spaces of product groups.

Provides insights into the algebraic and analytical properties of automorphism groups.

Abstract

Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $\Gamma$ with respect to $H$ admit a natural semigroup structure. For any compact group $K$ the semigroup $\Gamma$ acts in the space $L^2$ on the product of $m$ copies of $K$