Induced quasicocycles on groups with hyperbolically embedded subgroups

By Michael Hull, Denis V. Osin. 2012

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Let G be a group, H a hyperbolically embedded subgroup of G , V a normed G ‐module, U an H ‐invariant submodule of V . We propose a general construction which allows to extend 1‐quasicocycles on H with values in U to 1‐quasicocycles on G with values in V . As an application, we show that every group G with a nondegenerate hyperbolically embedded subgroup has dimH 2 b .G;‘ p .G//D1 for p 1. This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undistorted with respect to the stable commutator length.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/500947dfbb07a3ad234da5e91b6a1801479bfbc6

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Citations in the corpus

GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, background): “The latter theorem is especially useful in conjunction with various “extension” results proved in [1, 29, 40].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, background): “o [106]. It is known that H2 b(G;R) vanishes for amenable groups and all irreducible lattices in higher rank 1After the rst version of this paper was completed, M. Hull and the third author proved in [90] a more general extension theorem for quasi-cocycles, which also implies that dimH2 b (G;‘ p(G))”
Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “the relation between Creg, Dreg, and AH. First we recall that AH ⊆ Creg ∩Dreg. This was originally proved by Hamensta¨dt [43] in slightly diﬀerent terms and also follows from the results of Hull-Osin [50] and Bestvina–Bromberg–Fujiwara [17]. Moreover, acylindrically hyperbolic groups constitute the ”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “Acylindrical hyperbolicity has strong consequences: Every acylindrically hyperbolic group is SQ-universal, it contains free normal subgroups [DGO11], it contains Morse elements and hence all its asymptotic cones have cut-points [Sis16], and its bounded cohomology is infinite dimensional in degrees 2”
The median class and superrigidity of actions on CAT(0) cube complexes (2012, background): “ Γ is closed in the isometry group of X, [Ham]. Similarly, Hull and Osin proved that every group Γ with a suﬃciently nice hyperbolic subgroup has inﬁnite dimensional H2 b(Γ,ℓ p(Γ)), for 1 6 p < ∞, [HO]. Examples of groups satisfying such condition encompass, among others, groups Γ acting properly”