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Quasi-isometries between groups with two-ended splittings

Published online by Cambridge University Press:  30 June 2016

CHRISTOPHER H. CASHEN  and
ALEXANDRE MARTIN
CHRISTOPHER H. CASHEN
Affiliation:
Fakultät für Mathematik, Universität Wien Oskar-Morgenstern-Platz 1, 1090 Wien, Österreich. e-mail: christopher.cashen@univie.ac.at; alexandre.martin@univie.ac.at
ALEXANDRE MARTIN
Affiliation:
Fakultät für Mathematik, Universität Wien Oskar-Morgenstern-Platz 1, 1090 Wien, Österreich. e-mail: christopher.cashen@univie.ac.at; alexandre.martin@univie.ac.at
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Abstract

We construct a ‘structure invariant’ of a one-ended, finitely presented group that describes the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For hyperbolic groups satisfying a very general condition, these invariants completely reduce the problem of classifying such groups up to quasi-isometry to a relative quasi-isometry classification of the factors of their JSJ decomposition. Under some additional assumption, our results extend to more general finitely presented groups, yielding a far-reaching generalisation of the quasi-isometry classification of some 3–manifolds obtained by Behrstock and Neumann.

The same approach also allows us to obtain such a reduction for the problem of determining when two hyperbolic groups have homeomorphic Gromov boundaries.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 2 , March 2017 , pp. 249 - 291
Copyright
Copyright © Cambridge Philosophical Society 2016 

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