Groups, Geometry, and Dynamics

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Volume 4, Issue 3, 2010, pp. 473–516
DOI: 10.4171/GGD/92

Published online: 2010-06-16

Quasi-isometries between tubular groups

Christopher H. Cashen[1]

(1) University of Vienna, Wien, Austria

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.

Keywords: Tubular group, snowflake group, quasi-isometry

Cashen Christopher: Quasi-isometries between tubular groups. Groups Geom. Dyn. 4 (2010), 473-516. doi: 10.4171/GGD/92