Accidental Parabolics and Relatively Hyperbolic Groups

By constructing, in the relative case, objects analoguous to Rips and Sela’s canonical representatives, we prove that the set of conjugacy classes of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite. An important result of W.Thurston is : Theorem 0.1 ([T] 8.8.6) Let S be any hyperbolic surface of finite area, and N any geometrically finite hyperbolic 3manifold. There are only finitely many conjugacy classes of subgroups G ⊂ π1(N) isomorphic to π1(S) by an isomorphism which preserves parabolicity (in both directions). It is attractive to try to formulate a group-theoretic analogue of this statement : the problem is to find conditions such that the set of images of a groupG in a group Γ is finite up to conjugacy. If Γ is word-hyperbolic and G finitely presented, this has been the object of works by M.Gromov ([G] Theorem 5.3.C’) and by T.Delzant [Del], who proves the finiteness (up to conjugacy) of the set of images by morphisms not factorizing through an amalgamation or an HNN extension over a finite group. As a matter of fact, if a group G splits as A∗C B and maps to a group Γ such that the image of C in Γ has a large centralizer, then in general, there are infinitely many conjugacy classes of images of G in Γ. Technically speaking, if h is the considered map, one can conjugate h(A) by elements in the centralizer of h(C), without modifying h(B), hence producing new conjugacy classes of images. A similar phenomenon happens with HNN extensions. We are interested here in the images of a group in a relatively hyperbolic group (for example, a geometrically finite Kleinian group). Our result, Theorem 0.2, gives a condition similar to the one of Thurston, ruling out the bad situation depicted above, and ensuring the expected finiteness. Relatively hyperbolic groups were introduced by M.Gromov in [G], and studied by B.Farb [F] and B.Bowditch [B2], who gave different, but equivalent, definitions (see Definition 1.2 below, taken from [B2]). In Farb’s terminology, we are interested in “relatively hyperbolic groups with the property BCP”. The main example is the class of fundamental groups of geometrically finite manifolds (or orbifolds) with pinched negative curvature (see [B1], see also [F] for the case of finite volume manifolds). Sela’s limit groups are hyperbolic relative to their maximal abelian non-cyclic subgroups, as shown in [D]. Definition : We say that a morphism from a group in a relatively hyperbolic group h : G → Γ has an accidental parabolic either if h(G) is parabolic in Γ, or if h can be factorized through a ∗E-mail : [email protected]