An invitation to bounded cohomology

Abstract

Abstract.A selection of aspects of the theory of bounded cohomology is presented. The emphasis is on questions motivating the use of that theory as well as on some concrete issues suggested by its study. Specific topics include rigidity, bounds on characteristic classes, quasification, orbit equivalence, amenability. Mathematics Subject Classification (2000). Primary 55N; Secondary 20F.

Acylindrical hyperbolicity of groups acting on trees

“…Moreover, most known examples from these classes come from the world of groups acting on hyperbolic spaces. Motivated by these observations, Monod [66] and Thom [90] suggested to consider C reg and D reg as analytic analogues of the class of "negatively curved" groups. It is worth noting that no non-trivial relation between C reg and D reg is known, although they are likely to coincide (see Remark 2.7 …”

“…The first one was proved by Dahmani, Guirardel and the second author in [29 The next theorem was first proved in [43] under a certain assumption equivalent to acylindrical hyperbolicity (see [73]); it also follows from the main result of [50], where the language of hyperbolically embedded subgroups was used. For details about quasi-cocycles and bounded cohomology we refer to [43,50,66,90].…”

Groups not presentable by products

“…Finally, the non-vanishing of the second bounded cohomology with coefficients in the regular representation is a measure equivalence invariant [38], Corollary 7.6. Whether it is invariant under quasi-isometries seems to be unknown; see Problem J in Monod's 2006 ICM talk [36]. Table 1 surveys the applicability of different criteria to proving that certain classes of groups are not presentable by products.…”

Nonunitarizable Representations and Random Forests

“…A result of Gaboriau-Lyons [22], notably using [23] and [46], provides an F 2 -action on the Bernoulli percolation of any nonamenable countable group G in such a way that F 2 can be thought of as a "random subgroup" of G, even when G has no actual such subgroup. It was suggested in [38] (Problem N) to apply an induction procedure for specific representations of random subgroups in order to answer Dixmier's question. In fact, the first use of [22] toward a cohomological 4338 I. Epstein and N. Monod question asked in [24,Section 10] can be found in [38,Section 5] and the second use is the ergodic-theoretical result [17].…”

“…It was suggested in [38] (Problem N) to apply an induction procedure for specific representations of random subgroups in order to answer Dixmier's question. In fact, the first use of [22] toward a cohomological 4338 I. Epstein and N. Monod question asked in [24,Section 10] can be found in [38,Section 5] and the second use is the ergodic-theoretical result [17].…”