Quasi-isometries between groups with two-ended splittings

Christopher Cashen, Alexandre Martin

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
21 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)249-291
Number of pages43
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number2
Early online date30 Jun 2016
Publication statusPublished - Mar 2017


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