Let Out(Fn) denote the outer automorphism group of the free group Fn with n > 3. We prove that for any finite index subgroup Γ < Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian : every injective homomorphism Γ → Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).

This is the second in a series of papers about torsionfree groups which act properly and cocompactly on a CAT(0) metric space with isolated flats and relatively thin triangles. Our approach is to adapt the methods of Sela and others for wordhyperbolic groups to this context of non-positive curvature. The main result in this paper is that (under certain technical hypotheses) such a group as abov...

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We prove that the braid group on 4 or more strands modulo its center is co-Hopfian. We then show that any injective endomorphism of these braid groups is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from the braid group on n strands to the braid group on n + 1 strands is geometric (n ≥ 7). Additionally, we obtain related r...

This note investigates modules having quasi-injective and duo submodules. We introduce a new generalization of C_1-module. The main method that was adopted in this is how to obtain submodule N M the characteristic Quasi-injective. investigate relationship between pseudo-injective module Quasi-injective property Finally, we anti-hopfian module.