We give a method of constructing maps between tubular groups inductively according to a finite set of strategies. This map will be a quasi-isometry exactly when the set of strategies satisfies certain consistency criteria. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for building a quasi-isometry between them.
For two given tubular groups there are only finitely many candidate sets of strategies to consider, so it is possible in finite time to either produce a consistent set of strategies or decide that such a set does not exist. Consequently, there is an algorithm that in finite time decides whether or not two tubular groups are quasi-isometric.
Cite this article
Christopher H. Cashen, Quasi-isometries between tubular groups. Groups Geom. Dyn. 4 (2010), no. 3, pp. 473–516DOI 10.4171/GGD/92