Deciding isomorphy using Dehn fillings, the splitting case

• 成果类型: Article
• 署名作者: Dahmani, Francois; Touikan, Nicholas
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); University of New Brunswick
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0824-y
• 发表日期: 2019
• 页码: 81-169
• 关键词: relatively hyperbolic groups; finitely presented groups; general algorithms; geometry; graphs; deformation; SUBGROUPS; SPACES

摘要

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.