Extending higher-dimensional quasi-cocycles

Abstract

Abstract. Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(Fn) for n ≥ 2). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any R[G]-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.Bounded cohomology of discrete groups is very hard to compute. For example, as observed in [Mon06], there is not a single countable group G whose bounded cohomology (with trivial coefficients) is known in every degree, unless it is known to vanish in all positive degrees (this is the case, for example, of amenable groups). Even worse, no group G is known for which the supremum of the degrees n such that H n b (G, R) = 0 is positive and finite.In degree 2, bounded cohomology has been extensively studied via the analysis of quasi-morphisms (see e.g. [Bro81, EF97, Fuj98, BF02, Fuj00] for the case of trivial coefficients, and [HO13, BBF13] for more general coefficient modules). In this paper we exploit quasi-cocycles, which are the higherdimensional analogue of quasi-morphisms, to prove non-vanishing results for bounded cohomology in degree 3. To this aim, we prove that quasi-cocycles may be extended from a hyperbolically embedded family of subgroups to the ambient group. We also discuss the geometric information carried by our extensions of quasi-cocycles, showing in particular that projections on hyperbolically embedded subgroups may be reconstructed from the extensions of suitably chosen quasi-cocycles.Quasi-cocycles and bounded cohomology. Let G be a group, and V be a normed R[G]-space, i.e. a normed real vector space endowed with an

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

“…We can now proceed with the proof of Theorem 4, which states that r n+1 is an undistorted surjection for every n ≥ 1, provided that we endow both EH n+1 b (Γ) and EH n+1 b (H) with the · ∞,0 -seminorm (and that H n (H) is finitedimensional). The key ingredient for our argument will be an extension result for quasi-cocycles proved in [FPS15].…”

“…We first need to introduce the notion of small simplex in H. Such notion depends on the geometry of the embedding of H in Γ. However, for our purposes it is sufficient to know that we can single out a particular finite subset S 0 of H with the property that an element h ∈ H n+1 is small if and only if h ∈ S n+1 0 ⊆ H n+1 (see [FPS15,Definition 4.7]). In particular, the number of small simplices in H is finite, so for every cochain ϕ ∈ C n (H) the finite number…”

“…where B varies over all the left cosets of H in Γ, tr B n (g) is a sort of weighted projection of g into B (see [FPS15,Definition 4.5]), and ϕ ′ is obtained from ϕ via the left translation by an element of B (after setting ϕ = 0 on small simplices contained in H). It is proved in [FPS15, Theorem 5.1] that the sum in the above definition is finite (i.e.…”

“…First, we construct a discrete 3dimensional volume form on a class of free-by-cyclic groups. Then, building on results from [FPS15], we exploit our construction to show that, for every acylindrically hyperbolic group, the space of bounded classes with vanishing seminorm is infinite dimensional in degree 3.…”

Low-dimensional bounded cohomology and extensions of groups

“…If G is amenable then it is known that H n b (G, R) = 0 for all n ≥ 1. On the other hand if G is "non-positively curved" then H 2 b (G, R) and H 3 b (G, R) are typically infinite dimensional as an R-vectorspace, for example for acylindrically hyperbolic groups; see [11] and [5]. However, there is no full characterisation of all bounded classes in H n b (G, R) for n = 2, 3.…”