Residual finiteness of outer automorphism groups of relatively hyperbolic groups
Joint work with Gilbert Levitt and Denis Osin.
I like relatively hyperbolic groups, since combinatorial and geometric group theory meet there.
1. Residual finiteness
Definition 1 A group is residually finite if the intersection of all normal finite index subgroups of is trivial.
It is a useful property. Such groups have solvable word problem, they are Hopfian. Large classes of groups have it (polycyclic groups, linear groups,…). Some don’t.
Example: Baumslag-Solitar group is not residually finite.
Question: Are all word hyperbolic groups residually finite ?
Personnally, I think the answer must be no.
Question: When is residually finite ?
The case of is rather well understood.
Theorem 2 (Baumslag) If is finitely generated and residually finite, is residually finite.
The situation is opposite for .
Theorem 3 (Bumagina, Wise) Every finitely presented group is the outer automorphism group of some finitely generated and residually finite group.
So extra conditions are required.
1.2. Conjugacy separability
Definition 4 A group is conjugacy separable if non conjugate elements are still non conjugate in some finite quotient.
Definition 5 Say an automorphism of is pointwise inner if for every , is conjugate to . The group of pointwise automorphisms of is denoted by .
Theorem 6 (Grossman) If is finitely generated, conjugacy separable, and , then is residually finite.
Proof: Let . By assumption, there is such that is not conjugate to . In some finite quotient, is not conjugate to again. This implies that is non trivial in the outer automorphism group of that finite quotient.
Example: Virtually polycyclic groups, virtually free groups, virtual surface groups (A. Martino), limit groups are conjugacy separable.
Example: is not conjugacy separable.
Note that unlike residual finiteness, conjugacy separability does not pass to finite index subgroups or overgroups.
The property is easier to establish.
2. Relative hyperbolicity
Let be finitely generated by , let be subgrous of . Then is a quotient of . By a finite relative presentation, I mean that the quotient is normally generated by finitely many elements.
Definition 7 Choose as an alphabet
Assume has finite presentation with respect to the set of generators with set of relators . If is a word in this alphabet which is trivial in ,
define smallest possible as the filling area of .
Say that is hyperbolic relative to if there exists a constant such hat for every word in alphabet which is trivial in , .
Example: Limit groups are hyperbolic with respect to free abelian subgroups.
2.2. Non relatively hyperbolic groups
Definition 8 Say is NRH is is not hyperbolic relative to any family of proper subgroups.
Example: , , are NRH.
3.1. The one ended case
Theorem 9 (Levitt, Minasyan) Suppose that is finitely generated, one-ended and hyperbolic relative to . If are NRH and residually finite, then is residually finite.
Note that this does not assume nor imply that itself is residually finite.
Corollary 10 For every one-ended word hyperbolic group , is residually finite.
Note that embeds in . Since has infinitely many ends, one cannot conclude that , and thus , is residually finite. Too bad.
3.2. The infinite ended case
Theorem 11 (Minasyan, Osin) Suppose that is finitely generated, residually finite group with infinitely many ends, then is residually finite.
The infinite ended assumption implies that is hyperbolic relative to a family of subgroups.
We use graphs of groups.
Definition 12 Let be a graph of groups built on a finite, connected, bipartite graph . Assume that at right vertices,
- Vertex group is hyperbolic relative to where are the edges of adjacent to .
- There is a finite subset such that all finite index normal subgroups not intersecting satisfy . Furthermore, is conjugacy separable, where is the subgroup normally generated by the ‘s.
Here is our main technical result.
Theorem 13 (Levitt, Minasyan) In addition to the assumptions above on right vertex groups, assume that is finitely generated, does not coincide with a left vertex group, has no nontrivial finite normal subgroups. Assume that all left vertex groups are residually finite. Then is residually finite.
The second theorem (infinitely many ends) follows. According to Stallings, splits along a finite subgroup or is an HNN extension. So is a graph of groups with graph with one edge. Add a vertex in the middle of the edge. The assumptions on right vertices are trivially satisfied.
In the first theorem (one end), one uses the canonical JSJ decomposition of Guirardel and Levitt. Vertices split into 3 types : elementary, surface and rigid. Put all elementary vertices on the left, surface and rigid vertices on the right. Outer automorphism groups of rigid vertex groups relative to the
incoming edge groups are finite. So up to finite index, one can
replace rigid vertices by
free products of incoming edge groups with a free group.
The following are equivalent:
- Every word hyperbolic group is residually finite.
- For every word hyperbolic group, is residually finite.