Deciding Isomorphy using Dehn fillings, the splitting case
Abstract
We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.
Cite
@article{arxiv.1311.3937,
title = {Deciding Isomorphy using Dehn fillings, the splitting case},
author = {François Dahmani and Nicholas Touikan},
journal= {arXiv preprint arXiv:1311.3937},
year = {2020}
}
Comments
76 pages. This version incorporates referee comments and corrections. The main changes to the previous version are a better treatment of the algorithmic recognition and presentation of virtually cyclic subgroups and a new proof of a rigidity criterion obtained by passing to a torsion-free finite index subgroup. The previous proof relied on an incorrect result. To appear in Inventiones Mathematicae