Quasi-isometries Between Tubular Groups

TL;DR

This paper introduces an algorithmic method to determine quasi-isometries between tubular groups by constructing maps based on strategies, providing a decision procedure for their quasi-isometric classification.

Contribution

It develops a novel inductive approach and an algorithm to decide quasi-isometry between tubular groups, linking strategy consistency with geometric equivalence.

Findings

The method constructs quasi-isometries from consistent strategies.

An algorithm can decide the existence of a quasi-isometry in finite time.

The approach characterizes quasi-isometry through strategy consistency.

Abstract

We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric.