Endomorphisms of Kleinian groups

Abstract

Ag roupG is cohopfian (or has the co-Hopf property) if any injective endomorphism f : G → G is surjective. Answering a question of E. Rips, Z. Sela showed in [S2] that a torsionfree, non-virtually cyclic word-hyperbolic group (in Gromov’s sense) is cohopfian if and only if it is not a non-trivial free product. The cohopficity of 3-manifold groups has been studied by many authors; see [PW] and [OP] where a more complete list of references on this subject is given. A non-trivial free product A ∗ B is never cohopfian, as it contains the proper subgroup A∗mBm −1 isomorphic to A∗B if m/ ∈ (A∪B). More generally, let the group G split as an HNN-extension, G = A∗C = � A, t | tCt −1 = ϕ(C)� , and suppose that t centralizes C .T henG is not cohopfian (set f : G → G be the identity on A and f (t )= t 2 ;t henf is injective, not surjective). It is shown in [OP] that this example can be realized as a Kleinian group. Note that in this case, the group G splits over a parabolic subgroup C which is of infinite index in the unique maximal parabolic subgroup ˜