The second bounded cohomology of a group acting on aGromov-hyperbolic space

Abstract

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.