Bounded cohomology of finitely generated Kleinian groups

Abstract

Any action of a group Γ on H 3 by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to Γ. We prove that the bounded cohomology of finitely generated Kleinian groups without parabolic elements distinguishes the asymptotic geometry of geometrically infinite ends of hyperbolic 3manifolds. 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.

Transgression in bounded cohomology and a conjecture of Monod

“…[7,25,20,5,15,16,4,32,1,30]), and bounded cohomology in degree 3, which has close ties with the geometry of 3-manifolds (e.g. [7,46,47,48,21,22,17,45,23]). Bounded cohomology in higher degrees, on the contrary, is still largely unexplored.…”

Relations in bounded cohomology

“…because the Lipschitz constant cK bounds the Jacobian of id t by (cK) 2 , and the area of a hyperbolic triangle is no more than π. Combining estimates (8), (14) and 18 Next, we show that singly degenerate bounded fundamental classes can be represented by a class defined on the whole manifold S × R, but which has support only in the convex core. This will come in handy when we prove Theorem 1.2.…”

“…More precisely, by Lemma 7.1 and by construction of f − (see Remark 7.2), there is a D such that if p ∈ [i, j] ⊂ Δ 2 and q = Φ − * str − τ (p) ∈ supp(f − ), then d M (Π(q), q) D , where Π : H 3 − → str Φ − * τ ([i, j]) is closest point projection. Take c := (2B log√ with (9)-(14) in the proof of Proposition 3.5. Recall that Φ − is B-Lipschitz.…”

Borel and volume classes for dense representations of discrete groups

“…The quasi-isometry class of an injective Kleinian surface group representation ρ : Γ − → PSL 2 C without parabolic elements is characterized by the volume class [ρ * vol 3 ] ∈ H 3 b (Γ; R) [Far18a,Far18b]. The semi-norm in bounded cohomology can be used to detect faithfulness of arbitrary representations [Far18a,Theorem 7.8], and Theorem 1.1 says that the semi-norm on bounded cohomology detects discreteness for Zariski dense representations. We have the following rigidity property, which is a consequence of the work here and in [Far18a,Far18b].…”

“…The semi-norm in bounded cohomology can be used to detect faithfulness of arbitrary representations [Far18a,Theorem 7.8], and Theorem 1.1 says that the semi-norm on bounded cohomology detects discreteness for Zariski dense representations. We have the following rigidity property, which is a consequence of the work here and in [Far18a,Far18b]. We think of this as a kind of volume rigidity result.…”