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On canonical splittings of relatively hyperbolic groups

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Abstract

A JSJ decomposition of a group is a splitting that allows one to classify all possible splittings of the group over a certain family of edge groups. Although JSJ decompositions are not unique in general, Guirardel–Levitt have constructed a canonical JSJ decomposition, the tree of cylinders, which classifies splittings of relatively hyperbolic groups over elementary subgroups.

In this paper, we give a new topological construction of the Guirardel–Levitt tree of cylinders, and we show that this tree depends only on the homeomorphism type of the Bowditch boundary. Furthermore, the tree of cylinders admits a natural action by the group of homeomorphisms of the boundary. In particular, the quasi-isometry group of (G, ℙ) acts naturally on the tree of cylinders.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Binghamton University, P. O. Box 6000, Binghamton, NY, 13902, USA

    Matthew Haulmark

  2. Department of Mathematical Sciences, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, WI, 53201, USA

    G. Christopher Hruska

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  1. Matthew Haulmark
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  2. G. Christopher Hruska
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Correspondence to Matthew Haulmark.

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Haulmark, M., Hruska, G.C. On canonical splittings of relatively hyperbolic groups. Isr. J. Math. 258, 249–286 (2023). https://doi.org/10.1007/s11856-023-2472-1

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  • DOI: https://doi.org/10.1007/s11856-023-2472-1

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