Bounded cohomology of acylindrically hyperbolic groups
1. Bounded cohomology
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 1 A 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 .
Theorem 3 (Hull-Osin, Frigerio-Perretti-Sisto) If , then the restriction map
Corollary 4 If 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
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 6 If , 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.