## Notes of Alessandro Sisto’s Cambridge lecture 11-01-2017

Bounded cohomology of acylindrically hyperbolic groups

Joint with Frigerio, Pozzetti, Hardnik, Antolin, Mj, Taylor,…

1. Acylindrical hyperbolicity

Definition 1 (Osin) A group ${G}$ is acylindrically hyperbolic if it contains an infinite hyperbolically embedded subgroup of infinite index.

Let ${Y}$ be a possibily infinite generating system for ${G}$, such that ${Cay(G,Y)}$ is hyperbolic. Then ${H is hyperbolically embedded in ${(G,Y)}$ if

1. ${H}$ is quasi-isometrically embedded in ${Cay(G,Y)}$.
2. (Malnormality) Intersections of tubular neighborhoods of ${H}$ and its images by elements of ${G}$ have uniformly bounded diameter.

It turns out that ${H}$ is not unique, there is a wide choice. In fact, Maher and I showed that given 2 independant lazy simple random walks ${X_n}$ and ${Y_n}$, then with probability tending to 1, the subgroup ${\langle X_n,Y_n\rangle}$ is free and hyperbolically embedded.

2. Bounded cohomology

Theorem 2 (Hull-Osin, Franceschini-Frigerio-Pozzetti-Sisto) Let ${H}$ be hyperbolically embedded in ${(G,Y)}$. The restriction morphism from ${G}$ to ${H}$ in exact bounded cohomology is surjective.

In particular, any homogeneous quasimorphism on ${H}$ extends. It follows that ${EH^2_b(G)}$ and the torsion in ${EH^3_b(G)}$ are infinite dimensional.

Exact is necessary: group cohomology need not extend from ${H}$to ${G}$. For instance, for a manifold with boundary, the volume class of the boundary typically does not extend.

2.1. Random free subgroups

Lat ${\mathcal{A}}$ be a collection of hyperbolically embedded subgroups. To go further, we study the simultaneous restriction ${res_{\mathcal{A}}}$ in exact bounded cohomology to all groups in ${\mathcal{A}}$.

Theorem 3 (Hartnik-Sisto) Assume ${\mathcal{A}}$ consists of all hyperbolically embedded free rank 2 subgroups, ${res_{\mathcal{A}}}$ is injective.

Proof is probabilistic: a quasimorphism restricts nontrivially to ${\langle X_n,Y_n\rangle}$ with high probability.

2.2. Intersections of conjugates of hyperbolically embedded subgroups

Say a finite collection of quasimorphisms ${\phi_i}$ defined on subgroups ${H_i}$ is intersection compatible if each time ${x\in H_i}$ and ${y\in H_j}$ are conjugate, ${\phi_i(x)=\phi_j(y)}$.

Theorem 4 (Antolin-Mj-Sisto-Taylor) Intersection compatible quasimorphisms extend to a global quasimorphism. If furthermore ${H_i}$ have infinite index, then simultaneous restriction is not injective.