A smallest irreducible lattice in the product of trees

Janzen, David; Wise, Daniel T

doi:10.2140/agt.2009.9.2191

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

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 $X$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>X</mi></math>$ containing exactly four squares. Its universal cover $˜ X ≅ T_{4} \times T_{4}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mrow><mi>X</mi></mrow><mo class="MathClass-op">̃</mo></mover><mo class="MathClass-op">≅</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub> <mo class="MathClass-bin">×</mo> <msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub></math>$ is isomorphic to the product of two $4$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>$ –valent trees. The group $π_{1} X$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>X</mi></math>$ is a lattice in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Aut</mo><mrow><mo class="MathClass-open">(</mo><mrow><mover accent="true"><mrow><mi>X</mi></mrow><mo class="qopname">̃</mo></mover></mrow><mo class="MathClass-close">)</mo></mrow></math>$ but $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>X</mi></math>$ 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mrow><mi>X</mi></mrow><mo class="MathClass-op">̃</mo></mover></math>$ 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)