## Notes of Alessandro Sisto’s Rennes lecture

Bounded cohomology of acylindrically hyperbolic groups

1. Bounded cohomology

1.1. Definition

To define cohomology of a group ${G}$, use ${G}$-invariant functions on ${G^{n+1}}$ as ${n}$-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 ${G}$ be a lattice in ${PSL(2,{\mathbb R})}$ acting on hyperbolic plane. Let ${o}$ be an origin in ${H^2}$. Define

$\displaystyle \begin{array}{rcl} c(g_0,g_1,g_2)=\textrm{ signed area of triangle }\Delta(g_0o,g_1o,g_2o). \end{array}$

This defines a non trivial 2-cohomology class.

Definition 1 A quasimorphism if a function ${q}$ on ${G}$ such that ${|q(gh)-q(g)-q(h)|}$ is bounded.

Then ${q(g_0^{-1}g_1)}$ is a 1-cochain, and its coboundary is a bounded 2-cocycle. Conversely, all of ${EH^2_b(G):=\mathrm{Ker}(H^2_b(G)\rightarrow H^2(G))}$ arises in this way.

Example. On free group ${F_2}$, given a word ${w}$, define ${q(g)=}$ number of occurrences of ${w}$ in ${g}$ minus the number of occurrences of ${w}$ in ${g^{-1}}$. This is a quasimorphism.

1.2. What is it good for ?

Cyclic central extensions of ${G}$ are classified by ${H^2(G)}$. Bounded classes provide extensions which are quasiisometric to ${G\times{\mathbb Z}}$.

Ghys: ${H^2_b(G)}$ 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 ${H}$ is hyperbolically embedded in ${G}$, denoted by ${H\hookrightarrow_h G}$, if there exists an isometric action of ${G}$ on some hyperbolic metric space ${X}$ such that

1. ${H}$ is quasiconvex in ${X}$.
2. Tubular neighborhoods of orbits tend not to intersect much, i.e. for all ${g\notin H}$,

$\displaystyle \begin{array}{rcl} \mathrm{diameter}((H+R)\cap(gH+R))\leq K(R). \end{array}$

3. ${H}$ acts properly on ${X}$.

Say ${G}$ is acylindrically hyperbolic if it admits a virtually free hyperbolically embedded subgroup.

Philosophy. Acylindrically hyperbolic${\Rightarrow}$ SQ-universal, i.e. every countable group embeds into some quotient of ${G}$.

3. Results

Theorem 3 (Hull-Osin, Frigerio-Perretti-Sisto) If ${H\hookrightarrow_h G}$, then the restriction map

$\displaystyle \begin{array}{rcl} res:EH^n_b(G)\rightarrow EH^n_b(H) \end{array}$

is surjective.

Corollary 4 If ${G}$ is acylindrically hyperbolic, then ${H^2_b(G)}$ and ${H^3_b(G)}$ are infinite dimensional.

Theorem 5 (Hartnick-Sisto) Let ${G}$ be acylindrically hyperbolic. Let ${\mathcal{H}=\{H\hookrightarrow_h G\,;\,H}$ virtually ${F_2\}}$. Then the restriction map

$\displaystyle \begin{array}{rcl} res:H^2_b(G)\rightarrow \prod_{H\in\mathcal{H}}H^2_b(H) \end{array}$

is injective.

4. Proof

4.1. Scheme of proof of injectivity

Assume ${H\hookrightarrow_h G}$ is ${F_2}$ and that one wants to pull back a quasimorphism ${q:G\rightarrow{\mathbb R}}$.

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

Second step. Assume ${q}$ is homogeneous and is not a homomorphism. Then there exist ${g}$, ${h}$ such that ${\Delta(g,h)=q(gh)-q(g)-q(h)\not=0}$. It follows that with probability bounded below, ${\Delta(X_n,Y_n)\not=0}$ for all large enough ${n}$. Thus the induced homogeneous quasimorphism is nonzero, its cohomology class is nonzero.

4.2. Proof of second step

Lemma 6 If ${\Delta(g,h)\not=0}$, then one of the following is non zero: ${\Delta(gh,x)}$, ${\Delta(g,hx)}$, ${\Delta(h,x)}$.

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 ${G}$ is essentially contained in an orbit of ${H}$, but for thin parts that we can neglect.