Volume 9, issue 4 (2009)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
A smallest irreducible lattice in the product of trees

David Janzen and Daniel T Wise

Algebraic & Geometric Topology 9 (2009) 2191–2201
Abstract

We produce a nonpositively curved square complex X containing exactly four squares. Its universal cover X̃T4 × T4 is isomorphic to the product of two 4–valent trees. The group π1X is a lattice in Aut(X̃) but π1X 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 X̃ 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