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Last time: Theorem 21 (Groves–Manning–Osin): If is hyperbolic rel then there exists a finite subset such that if then
(a) is injective;
(b) is hyperbolic rel .
Theorem 22 (Gromov, Olshanshkii, Delzant): If is hyperbolic relative to the infinite cyclic then there is a such that for all there exists a hyperbolic such that for each .
The proof is an easy application of Groves–Manning–Osin.
Definition: If (infinite cyclic) is malnormal then we say are independent. A group G is omnipotent if for every independent there exists a such that for all there exists a homomorphism $\phi$ from to a finite group such that for all .
Omnipotence strengthens residual finiteness for torsionfree groups.
Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.
We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.
Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup of any hyperbolic group is separable.
Let . The idea is to Dehn fill to get a new hyperbolic group in which the image is finite and . If we could do this, we would be done by residual finiteness. This works if is malnormal. But it probably isn’t. Fortunately, we can quantify how far is from being malnormal:
Definition: The height of is the maximal such that there are distinct cosets such that the intersection
H is height iff is finite. In a torsionfree group, is height iff is malnormal.
Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.
Agol, Groves and Manning are able to prove:
Theorem 24: Let be a (torsionfree) residually finite hyperbolic group, and a quasiconvex subgroup of height . Let . Then is an epimorphism to a hyperbolic group such that
(i) is quasiconvex in ;
(iii) has height .
The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of . Theorem 22 is an easy consequence.
We still need to convince ourselves of some basic facts about the previous lecture, for example is the map injective?
Example: Cut the sphere along the equator. Then the diagram we have isDefinition: If or we say that G splits over C, and we call C the edge group. If or and is not or in the latter case, then we say that splits non-trivially.
Definition: Let be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex (resp. edge ) let (resp. ) be a group. If are vertices adjoining an edge e then let be an injective homomorphism. This data determines a graph of groups .
We say that has:
- underlying graph
- vertex groups
- edge groups
- edge maps
Similarly, we have:
Definition: Let be a connected graph. For each vertex (resp. ) let (resp. ) be a connected CW-complex. If adjoin let be -injective continuous maps. This data determines a graph of spaces . It has underlying graph , vertex spaces , edge spaces , etc. The graph of spaces determines a space as follows: define
where for . We say that is a graph-of-spaces structure (or decomposition) for .
Remark: There is a natural map (by collapsing all the edge and vertex spaces).
Given any graph of groups we can construct a graph of spaces with underlying graph by assigning and realizing the edge maps as continuous maps . We write for . This is well-defined up to homotopy equivalence.
Definition: The fundamental group of is just
- If then
- If then
- Let be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along decomposes into a graph of spaces and into a graph of groups.
Note: The edge maps of are only defined up to free (i.e. unbased) homotopy. Translated to , this means that only the conjugacy class of in matters.
Remark: The map induces a surjection
Here’s a way to construct a graph of groups. Let’s suppose acts on a tree without edge inversions (we can do this by subdividing edges if necessary). Let . The group acts diagonally on The quotient has a structure of a graph of spaces. The underlying graph is and there is a natural map .
Let be a vertex below . The preimage of is just where is the stabilizer of . Similarly, for below , the preimage of is
If adjoins then so the edge map is a covering map and therefore -injective. We have defined a graph of spaces and since is simply connected.
Applying to everything, we have a graph of groups . Its underlying graph is . Its vertex groups are the vertex stabilizers of , its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, .
Question for next time: Does every graph of groups arise in this way?