** 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.

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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.

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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.*

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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. *

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