Bounded cohomology of acylindrically hyperbolic groups
Joint with Frigerio, Pozzetti, Hardnik, Antolin, Mj, Taylor,…
1. Acylindrical hyperbolicity
Definition 1 (Osin) A group is acylindrically hyperbolic if it contains an infinite hyperbolically embedded subgroup of infinite index.
Let be a possibily infinite generating system for , such that is hyperbolic. Then is hyperbolically embedded in if
- is quasi-isometrically embedded in .
- (Malnormality) Intersections of tubular neighborhoods of and its images by elements of have uniformly bounded diameter.
It turns out that is not unique, there is a wide choice. In fact, Maher and I showed that given 2 independant lazy simple random walks and , then with probability tending to 1, the subgroup is free and hyperbolically embedded.
2. Bounded cohomology
Theorem 2 (Hull-Osin, Franceschini-Frigerio-Pozzetti-Sisto) Let be hyperbolically embedded in . The restriction morphism from to in exact bounded cohomology is surjective.
In particular, any homogeneous quasimorphism on extends. It follows that and the torsion in are infinite dimensional.
Exact is necessary: group cohomology need not extend from to . For instance, for a manifold with boundary, the volume class of the boundary typically does not extend.
2.1. Random free subgroups
Lat be a collection of hyperbolically embedded subgroups. To go further, we study the simultaneous restriction in exact bounded cohomology to all groups in .
Theorem 3 (Hartnik-Sisto) Assume consists of all hyperbolically embedded free rank 2 subgroups, is injective.
Proof is probabilistic: a quasimorphism restricts nontrivially to with high probability.
2.2. Intersections of conjugates of hyperbolically embedded subgroups
Say a finite collection of quasimorphisms defined on subgroups is intersection compatible if each time and are conjugate, .
Theorem 4 (Antolin-Mj-Sisto-Taylor) Intersection compatible quasimorphisms extend to a global quasimorphism. If furthermore have infinite index, then simultaneous restriction is not injective.