I would like to ask a question about automorphisms of free products of groups.
More specifically, let $G = G\_1 \ast ... \ G\_n \ast F\_r$ where $F\_r$ is free group on r generators. We can define the group $Out(G)$ of outer automorphisms of $G$, as the quotient of automorphisms with the inner automorphisms.


Then we can see as a subgroup of $Out(G)$, the automorphisms that have a representative $\phi$ from $G$ to $G$, which induce the identity on $G\_i$'s and restricted to $F\_r$ is an automorphism from $F\_r$ to $F\_r$.
My question is if this subgroup shares any properties with $Out(F\_r)$, for example can we say that this subgroup is virtually torsion free or maybe every torsion subgroup is finite?


Thanks a lot in advance for your time.



A subgroup of outer automorphisms group of a free product
**EDITED:** Thanks to @Mathieu for finding a bug.


The group you describe is actually isomorphic to $\text{Aut}(F\_r)$ (which, of course, has a well known relation to $\text{Out}(F\_r)$). 


To see why, you have defined a subgroup of $\text{Aut}(G)$ which I will denote $A\_0$, consisting of all automorhisms which take each $G\_i$ to itself by the identity and which take $F\_r$ to itself by some automorphism. Restricting each element of $A\_0$ to $F\_r$ defines an isomorphism $A\_0 \mapsto \text{Aut}(F\_r)$, so we just need to show that the induced map $A\_0 \mapsto \text{Out}(G)$ is injective. For this it suffices to show that for each $g \in G$ with corresponding inner automorphism $i\_g(h) = ghg^{-1}$, and for each $\phi \in A\_0$, if $i\_g \circ h \in A\_0$ then $g$ is the identity. 


For each nontrivial free factor $B \in \{G\_1,...,G\_n,F\_r\}$, if $g \not\in B$ then for any non-identity element $h \in B$ we have $ghg^{-1} \not\in B$, contradicting invariance of $B$. Therefore $g$ is an element of each of the free factors $B$, and this forces $g$ to be the identity (assuming that at least one of the $G\_i$'s is nontrivial and that $r \ge 1$).


