Deciding isomorphy using Dehn fillings, the splitting case

François Dahmani, Nicholas Matheson Toui-kan

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)81-169
Number of pages89
JournalInventiones Mathematicae
Volume215
Issue number1
DOIs
StatePublished - Jan 17 2019

Fingerprint

Dehn Filling
Isomorphism Problem
Parabolic Subgroup
Automorphism Group
Relatively Hyperbolic Groups
Outer Automorphism Groups
Graph of Groups
Canonical Decomposition
Finitely Generated Group
Nilpotent Group
Torsion-free
Congruence
Torsion
Isomorphism
Isomorphic
Orbit
Vertex of a graph

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Dahmani, François ; Matheson Toui-kan, Nicholas. / Deciding isomorphy using Dehn fillings, the splitting case. In: Inventiones Mathematicae. 2019 ; Vol. 215, No. 1. pp. 81-169.
@article{ca926c922d0442bf8df02dcc8d361571,
title = "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.",
author = "Fran{\cc}ois Dahmani and {Matheson Toui-kan}, Nicholas",
year = "2019",
month = "1",
day = "17",
doi = "https://doi.org/10.1007/s00222-018-0824-y",
language = "English (US)",
volume = "215",
pages = "81--169",
journal = "Inventiones Mathematicae",
issn = "0020-9910",
publisher = "Springer New York",
number = "1",

}

Deciding isomorphy using Dehn fillings, the splitting case. / Dahmani, François; Matheson Toui-kan, Nicholas.

In: Inventiones Mathematicae, Vol. 215, No. 1, 17.01.2019, p. 81-169.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Deciding isomorphy using Dehn fillings, the splitting case

AU - Dahmani, François

AU - Matheson Toui-kan, Nicholas

PY - 2019/1/17

Y1 - 2019/1/17

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85054155578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054155578&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s00222-018-0824-y

DO - https://doi.org/10.1007/s00222-018-0824-y

M3 - Article

VL - 215

SP - 81

EP - 169

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -