Bounded cohomology of finitely generated Kleinian groups

Abstract

Any action of a group $\Gamma$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $\Gamma$. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic $3$-manifolds. That is, if two homotopy equivalent hyperbolic manifolds with infinite volume and without parabolic cusps have different geometrically infinite end invariants, then they define a $2$ dimensional subspace of bounded cohomology. Our techniques apply to classes of hyperbolic $3$-manifolds that have sufficiently different end invariants, and we give explicit bases for vector subspaces whose dimension is uncountable. We also show that these bases are uniformly separated in pseudo-norm, extending results of Soma. The technical machinery of the Ending Lamination Theorem allows us to analyze the geometrically infinite ends of hyperbolic $3$-manifolds with unbounded geometry.