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).

A module M is called epi-retractable if every submodule of M is a homomorphic image of M. Dually, a module M is called co-epi-retractable if it contains a copy of each of its factor modules. In special case, a ring R is called co-pli (resp. co-pri) if RR (resp. RR) is co-epi-retractable. It is proved that if R is a left principal right duo ring, then every left ideal of R is an epi-retractable ...

We give simple examples of finitely presented Kazhdan groups with infinite outer automorphism groups, as arithmetic lattices in Lie groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answer...

We give simple examples of finitely presented Kazhdan groups with infinite outer automorphism groups, as arithmetic lattices in Lie groups. This answers a question of Paulin, independently answered by Ollivier and Wise by completely different methods. We also use results of Abels about compact presentability of p-adic groups to exhibit a finitely presented non-Hopfian Kazhdan group. This answer...

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups that admit a descending chain of normal finite-index subgroups, each which G. prove up finite index, these are always obtained by pulling back from free abelian quotient. give two applications: First, we show characteristic subgroup arises the ab...

Let p : M → B be a proper surjective map defined on an (n+ 2)-manifold such that each point-preimage is a copy of a hopfian n-manifold. Then we show that p is an approximate fibration over some dense open subset O of the mod 2 continuity set C′ and C′ \O is locally finite. As an application, we show that a hopfian n-manifold N is a codimension-2 fibrator if χ(N) 6= 0 or H1(N) ∼= Z2.

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples surfaces without boundary whose (pure) are not co-Hopfian; these first such endomorphisms that fail to be surjective. then prove that, subject some topological conditions on the domain surface, any continuous homomorphism (arbitrary) sends Dehn twists is i...