On canonical splittings of relatively hyperbolic groups

2019/12/02 by Matthew Haulmark, Haulmark, Matthew, G. Christopher Hruska +1 · 1 citation
Mathematics · #20F65 #20F67 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1912.00886

openalex publication_date 2019/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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