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<G} 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.


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