## Notes of François Dahmani’s lecture

Dehn fillings, congruence subgroups and the isomorphism problem for some relatively hyperbolic group

joint with Guirardel and Touikan.

Goal: solve the isomorphism problem for toral relatively hyperbolic groups.

Example 1 Fundamental groups of cusped negatively curved manifolds.

More examples arise from

– Free products.

– Amalgamations over maximal cyclic subgroups which are not parabolic.

– Amalgamations over a parabolic subgroups which is maximal in one of the summands.

I am interested in groups relatively hyperbolic with respect to nilpotent subgroups. This is the first class where techniques used previously (Sela, Dahmani, Guirardel) must fail. Indeed, earlier work relies on solving equations in groups. But solving equations in certain nilpotent groups is undecidable (Romankov). However, the isomorphism problem for the class of nilpotent groups is solvable (Grünewald-Segal).

Theorem 1 The isomorphism problem is solvable for the class of relatively hyperbolic groups with nilpotent parabolic subgroups.

0.1. Tools

1. We use the existence of congruence subgroups separating the torsion in nilpotent groups. Indeed, if ${P}$ is finitely generated nilpotent, there exists a characteristic subgroup ${N}$ of ${P}$, of finite index, such that the kernel of ${Out(P) \rightarrow Out(P/N)}$ is torsion free.

2. We use the solution of the mixed Whitehead in nilpotent groups. The mixed Whitehead problem is two decide wether two tuples of conjugacy classes differ by an automorphism. For nilpotent groups, this follows from arithmeticity of ${P}$ and ${Aut(P)}$ and a result of Grünewald and Segal.

0.2. Our strategy

– Cut RHG in pieces: JSJ decomposition (Guirardel-Levitt). The decomposition has elementary edge groups and maximal elementary, rigid or surface vertex groups. Touikan showed that in the torsion free case, the JSJ splitting is computable algorithmically.

– The isomorphism problem for rigid type vertices is solvable.

– The main difficulty has to do with the embeddings of edge groups in vertex groups. A graph of groups may be twisted by precomposing attaching maps with automorphisms of edge groups. We cannot decide wether such automorphisms extend to automorphisms of vertex groups. However, this works provided we pass to torsion separating congruence quotients.