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arXiv · 1912.00886

On canonical splittings of relatively hyperbolic groups

Matthew Haulmark·G. Christopher Hruska
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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 $(G,\mathbb{P})$ acts naturally on the tree of cylinders.

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Matthew Haulmark, G. Christopher Hruska. 2019-12-02. On canonical splittings of relatively hyperbolic groups. https://arxiv.org/abs/1912.00886

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