Extending group actions on metric spaces
Joint work with David Hume and C. Abbott.
Question. Let be groups. Given an isometric action of on a metric space , 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 act on and . Say that a map is coarsely equivariant if for every , is bounded on .
Definition 1 Say an action of group on is an extension of the action of subgroup on is there exists a coarsely -equivariant quasi-isometric embedding .
Definition 2 We say that the extension problem (EP) for is solvable if every action of on a metric space extends to an action of .
This is rather flexible.
- If has bounded orbits, the trivial action of is an extension.
- If is a retract of (i.e. there exists a homomorphism which is the identity on ), then every actions of extends.
- Fix finite generating systems of and . Assume is undistorted in . Then the action of on its Cayley graph extends to the action of on its Cayley graph.
- An example where (EP) is not solvable. Let . Then every action of on a metric space has bounded orbits (Cornulier). If , no action of with unbounded orbits can extend.
- A converse of (3) holds: if is finitely generated and (EP) is solvable for then is finitely generated and undistorted in . Whence many examples where (EP) is not solvable. Furthermore, if is finitely generated and elementarily amenable, then (EP) is solvable for all implies that is virtually abelian.
- Let be a free group and where exchanges generators. Then translation action of on with one generator acting trivially cannot extend to . Indeed, one generator of has bounded orbits, the other does not, but both are conjugate in .
1.2. Hyperbolic embeddings
The following definition appears in Dahmani-Guirardel-Osin. Let be a subset such that generates . Let be the metric on induced by the embedding of (as vertex set of complete graph ) into with edges of removed. Say that is hyperbolically embedded in if
- is hyperbolic,
- is proper.
- is not hyperbolically embedded into , but it is into .
- Observe that there exists a finite subset such that is hyperbolically embedded into iff is hyperbolic relative to .
- If is pseudo-Anosov, then there exists a virtually cyclic subgroup containing which is hyperbolically embedded in .
1.3. Acylindrically hyperbolic groups
This class contains , , finitely presented groups of deficiency (argument uses -Betti numbers).
Theorem 3 (Dahmani-Guirardel-Osin) If is acylindrically hyperbolic, then it contains hyperbolically embedded subgroups of the form finite for all .
Theorem 4 Let be hyperbolically embedded. Then (EP) is solvable for . Moreover, every action of on a hyperbolic metric space extends to a action of on a hyperbolic metric space.
Corollary 5 Let be a hyperbolic group, and .
- If is virtually cyclic, then (EP) is solvable for .
- If is quasi-convex and almost malnormal ( for all ), then (EP) is solvable for .
- Conversely, if (EP) for is solvable, then is quasi-convex.