We show continuity under equivariant Gromov–Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.

We present a careful approximation of the quasi-geodesics of trees of hyperbolic and relatively hyperbolic spaces. As an application we prove a dynamical and geometric combination theorem for trees of relatively hyperbolic spaces, with both Farb's and Gromov's definitions.

Suppose a group $G$ acts on a Gromov-hyperbolic space $X$ properly discontinuously. If the limit set $L(G)$ of the action has at least three points, then the second bounded cohomology group of $G$,$H^2_b(G;{\Bbb R})$ is infinite dimensional. For example, if $M$ is a complete, pinched negatively curved Riemannian manifold with finite volume, then $H_b^2(\pi _1(M); {\Bbb R})$ is infinite dimensional. As an application, we show that if $G$ is a knot group with $G \not\simeq{\Bbb Z}$, then $H^2_b(G;{\Bbb R})$ is infinite dimensional.

1991 Mathematics Subject Classification: primary 20F32; secondary 53C20, 57M25.