Let G be a group acting on a tree X. We show that some classical results concerning finitely generated subgroups of free groups, free products, and free-by-finite groups, remain valid if we replace finitely generated subgroups by tame subgroups or by subgroups of finite complexity. We also prove new results for tame subgroups and, more generally, for subgroups of finite complexity.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.
Abstract. It is shown, for a given graph group G, that the fixed point subgroup Fix ' is finitely generated for every endomorphism ' of G if and only if G is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms if the dependence graph of G is a transitive forest.
a b s t r a c tWe show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
Let G be the fundamental group of a graph of groups with finite edge groups and f an endomorphism of G. We prove a structure theorem for the subgroup Fix(f), which consists of the elements of G fixed by f, in the case where the endomorphism f of G maps vertex groups into conjugates of themselves.
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