A smallest irreducible lattice in the product of trees

Janzen, David; Wise, Daniel T

doi:10.2140/agt.2009.9.2191

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A smallest irreducible lattice in the product of trees

David Janzen and Daniel T Wise

Algebraic & Geometric Topology 9 (2009) 2191–2201

DOI: 10.2140/agt.2009.9.2191

Abstract

We produce a nonpositively curved square complex $X$ containing exactly four squares. Its universal cover $\tilde{X} ≅ T_{4} \times T_{4}$ is isomorphic to the product of two $4$ –valent trees. The group $π_{1} X$ is a lattice in $Aut (\tilde{X})$ but $π_{1} X$ is not virtually a nontrivial product of free groups. There is no such example with fewer than four squares. The main ingredient in our analysis is that $\tilde{X}$ contains an “anti-torus” which is a certain aperiodically tiled plane.

Keywords

irreducible lattice, CAT(0) cube complex

Mathematical Subject Classification 2000

Primary: 20F67

References

Publication

Received: 4 April 2007

Revised: 9 March 2009

Accepted: 26 August 2009

Published: 27 October 2009

Authors

David Janzen
Math & Stats McGill University Montreal, Quebec H3A 2K6 Canada

Daniel T Wise
Math & Stats McGill University Montreal, Quebec H3A 2K6 Canada
http://www.math.mcgill.ca/wise

Volume 9, issue 4 (2009)