Several remarks on groups of automorphisms of free groups

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 $K$