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<G} 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<G} 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<G}, 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<G} 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<G} 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<G} be hyperbolically embedded. Then (EP) is solvable for {H<G}. 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<G}.

  1. If {H} is virtually cyclic, then (EP) is solvable for {H<G}.
  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<G}.
  3. Conversely, if (EP) for {H<G} 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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