Trees of cylinders and canonical splittings

TL;DR

This paper introduces a method to construct canonical trees of cylinders from group actions on trees, yielding invariant JSJ splittings with strong compatibility properties, advancing the understanding of group decompositions.

Contribution

It defines trees of cylinders based on edge stabilizer equivalences, providing invariant and compatible JSJ splittings for finitely generated groups.

Findings

Constructed trees of cylinders depend only on the deformation space.

Obtained Out(G)-invariant cyclic or abelian JSJ splittings.

Trees of cylinders exhibit strong compatibility properties.

Abstract

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a common refinement).