Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended JSJ Decompositions

TL;DR

This paper introduces new quasi-isometry invariants for one-ended finitely presented groups using trees of cylinders from JSJ decompositions, with refined invariants under additional rigidity conditions.

Contribution

It develops a novel method to derive quasi-isometry invariants from the tree of cylinders in two-ended JSJ decompositions, including finer invariants under rigidity assumptions.

Findings

Constructed invariants distinguish groups up to quasi-isometry.

Finer invariants capture stretching across edges under rigidity conditions.

Applicable to classifying groups with specific JSJ decomposition structures.

Abstract

We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders.