** Bounded cohomology of acylindrically hyperbolic groups **

**1. Bounded cohomology **

** 1.1. Definition **

To define cohomology of a group , use -invariant functions on as -cochains, with the obvious coboundary. If one sticks to bounded functions, one gets bounded cohomology. There is an obvious map to usual cohomology.

**Example**. Let be a lattice in acting on hyperbolic plane. Let be an origin in . Define

This defines a non trivial 2-cohomology class.

Definition 1A quasimorphism if a function on such that is bounded.

Then is a 1-cochain, and its coboundary is a bounded 2-cocycle. Conversely, all of arises in this way.

**Example**. On free group , given a word , define number of occurrences of in minus the number of occurrences of in . This is a quasimorphism.

** 1.2. What is it good for ? **

Cyclic central extensions of are classified by . Bounded classes provide extensions which are quasiisometric to .

Ghys: classifies circle actions up to semiconjugacy.

There is a proof of Mostow’s rigidity theorem tat relies on bounded cohomology.

Several rigidity theorems concerning higher rank lattices follow from vanishing of their bounded cohomology.

**2. Acylindrically hyperbolic groups **

These are the groups which admit an isometric action on a Gromov hyperbolc space acylindrically. I prefer the following version of the definition.

Definition 2 (Dahmani-Guirardel-Osin)Say is hyperbolically embedded in , denoted by , if there exists an isometric action of on some hyperbolic metric space such that

- is quasiconvex in .
- Tubular neighborhoods of orbits tend not to intersect much, i.e. for all ,
- acts properly on .

Say is acylindrically hyperbolic if it admits a virtually free hyperbolically embedded subgroup.

**Philosophy**. Acylindrically hyperbolic SQ-universal, i.e. every countable group embeds into some quotient of .

**3. Results **

Theorem 3 (Hull-Osin, Frigerio-Perretti-Sisto)If , then the restriction map

is surjective.

Corollary 4If is acylindrically hyperbolic, then and are infinite dimensional.

Indeed, this is true for free groups.

Theorem 5 (Hartnick-Sisto)Let be acylindrically hyperbolic. Let virtually . Then the restriction map

is injective.

**4. Proof **

** 4.1. Scheme of proof of injectivity **

Assume is and that one wants to pull back a quasimorphism .

**First step**. Launch two independent random walks and on . As tends to infinity, the probability that the group generated by and be hyperbolically embedded tend to 1 (joint work with Joseph Maher).

**Second step**. Assume is homogeneous and is not a homomorphism. Then there exist , such that . It follows that with probability bounded below, for all large enough . Thus the induced homogeneous quasimorphism is nonzero, its cohomology class is nonzero.

** 4.2. Proof of second step **

Lemma 6If , then one of the following is non zero: , , .

Indeed, these are the 4 faces of a 3-simplex.

** 4.3. Proof of surjectivity **

Due to thinness of triangles, every triangle in an orbit of is essentially contained in an orbit of , but for thin parts that we can neglect.