Geodesic
Several remarks on groups of automorphisms of free groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXV, Tome 436 (2015), pp. 189-198

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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 $K$. 
@article{ZNSL_2015_436_a10,
     author = {Yu. A. Neretin},
     title = {Several remarks on groups of automorphisms of free groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {189--198},
     publisher = {mathdoc},
     volume = {436},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_436_a10/}
}
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Yu. A. Neretin. Several remarks on groups of automorphisms of free groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXV, Tome 436 (2015), pp. 189-198. http://geodesic.mathdoc.fr/item/ZNSL_2015_436_a10/