In this paper and its sequel, we prove that injective homomorphisms between Teichmüller modular groups of compact orientable surfaces are necessarily isomorphisms, if an appropriately measured “size” of the surfaces in question differs by at most one. In particular, we establish the co-Hopfian property for modular groups of surfaces of positive genus.

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using this characterization the related topological properties of 2-cuts, we prove that every necklace has a unique IFS in certain sense. Also, two necklaces admit only rigid homeomorphisms thus group self-homeomorphisms is countable. In addition, weaker co-Hopfian property also obtained.

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...

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...