Bounded cohomology and isometry groups of hyperbolic spaces

By U. Hamenstaedt. 2005

96 citations recorded by the source corpus.

Let $X$ be an arbitrary hyperbolic geodesic metric space and let $\Gamma$ be a countable subgroup of the isometry group ${\rm Iso}(X)$ of $X$. We show that if $\Gamma$ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups $H_b^2(\Gamma,\mathbb{R})$, $H_b^2(\Gamma,\ell^p(\Gamma))$ $(1< p <\infty)$ are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct product.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/daacadccdbb0c3a2a8e0323517174faeee48d308

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, background): “Let us comment on Hamenstädt’s recent work in [38].”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “This definition of Osin [41] unified several equivalent far-reaching generalizations of relative hyperbolicity [12, 17, 30, 44].”
Statistically convex-cocompact actions of groups with contracting elements (2016, background): “Thus, they all belong to the category of acylindrically hyperbolic groups, which are studied previously in various guises in [12], [11], [48], and [27] and in a continually growing body of literature.”
Quasi-morphisms on surface diffeomorphism groups (2019, background): “ly 80s [Gro82]. In the context of mapping class groups there are now many constructions of quasi-morphisms, starting with the work of Endo{Kotschick [EK01], Bestvina{ Fujiwara [BF02], and Hamenstadt [Ham08] to name only a few. Using the natural projection of the full dieomorphism group to the mappin”
Induced quasicocycles on groups with hyperbolically embedded subgroups (2012, background): “For example, this was known for hyperbolic groups (Mineyev, Monod and Shalom [17]) and more generally groups acting nonelementary and acylindrically on hyperbolic spaces (Hamenstädt [14]), groups acting properly on proper CAT.”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “For details about quasi-cocycles and bounded cohomology we refer to [43,50,64,88].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, result): “Indeed every group admitting a non-elementary acylindrical action on a hyperbolic space is in Creg by a result of Hamelshtadt [68].”
Stable commutator length in word-hyperbolic groups (2006, methodology): “Hamenstädt ([24]) used dynamical methods to directly construct quasimorphisms on word-hyperbolic groups.”
Measure equivalence rigidity of the mapping class group (2006, background): “In the proof of Furman’s rigidity theorem in [11], the main ingredient is to prove the following (see [11, Theorem 4.1]): let (Ω, ω) be a self ME coupling of Γ (i.e., a ME coupling of Γ and Γ).”
GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, background): “The following theorem was proved in several papers under various assumptions (see [8, 10, 36, 40] and references therein), which later turned out to be equivalent to acylindrical hyperbolicity.”
Acylindrically hyperbolic groups (2013, methodology): “One example is the weak acylindricity in the sense of Hamenst¨adt [25], which is weaker than acylindricity but stronger than weak proper discontinuity.”