Several remarks on groups of automorphisms of free groups

Let $\mathbb G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $\mathbb H$ be the stabilizer of the first $m$ generators of $F_\infty$. We show that the double cosets $\Gamma_m=\mathbb{H\setminus G/H}$ admit a natural semigroup structure. For any compact group $K$, the semigroup $\Gamma_m$ acts in the space $L^2$ on the product of $m$ copies of $K$.