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.

36 pages, 6 figures, V2 includes a much stronger version of a main result concerning the uniform separation of bounded fundamental classes in pseudonorm, V3 incorporates referee suggestions, V4 minor changes. To appear in GAFA

References in corpus (2)

• Existence of ruled wrappings in hyperbolic 3-manifolds
• The augmented marking complex of a surface

Cited by in corpus (2)

• Relations in Bounded Cohomology
• Borel and volume classes for dense representations of discrete groups