## Notes of Denis Osin’s Cambridge lecture 22-06-2017

Extending group actions on metric spaces

Joint work with David Hume and C. Abbott.

Question. Let ${H be groups. Given an isometric action of ${H}$ on a metric space ${S}$, does it extend to an action on a (possibly different) metric space ?

1. Extensions of actions

What to be mean by extension? We have in mind induction of representations.

Let ${H}$ act on ${S}$ and ${R}$. Say that a map ${f:S\rightarrow R}$ is coarsely equivariant if for every ${x\in R}$, ${h\mapsto d(f(hx),hf(x))}$ is bounded on ${H}$.

Definition 1 Say an action of group ${G}$ on ${R}$ is an extension of the action of subgroup ${H}$ on ${S}$ is there exists a coarsely ${H}$-equivariant quasi-isometric embedding ${f:S\rightarrow R}$.

Definition 2 We say that the extension problem (EP) for ${H is solvable if every action of ${H}$ on a metric space extends to an action of ${G}$.

1.1. Examples

This is rather flexible.

1. If ${H}$ has bounded orbits, the trivial action of ${G}$ is an extension.
2. If ${H}$ is a retract of ${G}$ (i.e. there exists a homomorphism ${G\rightarrow H}$ which is the identity on ${H}$), then every actions of ${H}$ extends.
3. Fix finite generating systems of ${H}$ and ${G}$. Assume ${H}$ is undistorted in ${G}$. Then the action of ${H}$ on its Cayley graph extends to the action of ${G}$ on its Cayley graph.
4. An example where (EP) is not solvable. Let ${G=Sym({\mathbb N})}$. Then every action of ${G}$ on a metric space has bounded orbits (Cornulier). If ${H, no action of ${H}$ with unbounded orbits can extend.
5. A converse of (3) holds: if ${G}$ is finitely generated and (EP) is solvable for ${H then ${H}$ is finitely generated and undistorted in ${G}$. Whence many examples where (EP) is not solvable. Furthermore, if ${G}$ is finitely generated and elementarily amenable, then (EP) is solvable for all ${H implies that ${G}$ is virtually abelian.
6. Let ${H=F_2}$ be a free group and ${G=H *_\phi}$ where ${\phi}$ exchanges generators. Then translation action of ${H}$ on ${{\mathbb R}}$ with one generator acting trivially cannot extend to ${G}$. Indeed, one generator of ${H}$ has bounded orbits, the other does not, but both are conjugate in ${G}$.

1.2. Hyperbolic embeddings

The following definition appears in Dahmani-Guirardel-Osin. Let ${X\subset G}$ be a subset such that ${X\cup H}$ generates ${G}$. Let ${\hat d}$ be the metric on ${H}$ induced by the embedding of ${H}$ (as vertex set of complete graph ${Cay(H,H)}$) into ${Cay(G,X\cup H)}$ with edges of ${Cay(H,H)}$ removed. Say that ${H}$ is hyperbolically embedded in ${(G,X)}$ if

1. ${Cay(G,X\cup H)}$ is hyperbolic,
2. ${(H,\hat d)}$ is proper.

For instance,

1. ${H}$ is not hyperbolically embedded into ${H\times{\mathbb Z}}$ , but it is into ${H*{\mathbb Z}}$.
2. Observe that there exists a finite subset ${X\subset G}$ such that ${H}$ is hyperbolically embedded into ${(G,X)}$ iff ${G}$ is hyperbolic relative to ${H}$.
3. If ${a\in MCG}$ is pseudo-Anosov, then there exists a virtually cyclic subgroup ${E}$ containing ${a}$ which is hyperbolically embedded in ${MCG}$.

1.3. Acylindrically hyperbolic groups

This class contains ${MCG}$, ${Out(F_n)}$, finitely presented groups of deficiency ${\geq 2}$ (argument uses ${\ell^2}$-Betti numbers).

Theorem 3 (Dahmani-Guirardel-Osin) If ${G}$ is acylindrically hyperbolic, then it contains hyperbolically embedded subgroups of the form ${F_n\times}$ finite for all ${n}$.

2. Results

Theorem 4 Let ${H be hyperbolically embedded. Then (EP) is solvable for ${H. Moreover, every action of ${H}$ on a hyperbolic metric space extends to a action of ${G}$ on a hyperbolic metric space.

Corollary 5 Let ${G}$ be a hyperbolic group, and ${H.

1. If ${H}$ is virtually cyclic, then (EP) is solvable for ${H.
2. If ${H}$ is quasi-convex and almost malnormal (${|H\cap H^g|<\infty}$ for all ${g\notin H}$), then (EP) is solvable for ${H.
3. Conversely, if (EP) for ${H is solvable, then ${H}$ is quasi-convex.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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