Geodesic
A smallest irreducible lattice in the product of trees
Janzen, David1 ; Wise, Daniel T1
1Math & Stats, McGill University, Montreal, Quebec H3A 2K6, Canada
Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2191-2201
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We produce a nonpositively curved square complex X containing exactly four squares. Its universal cover X̃≅T4 × 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.

DOI : 10.2140/agt.2009.9.2191
Keywords: irreducible lattice, CAT(0) cube complex

Janzen, David  1   ; Wise, Daniel T  1

1 Math & Stats, McGill University, Montreal, Quebec H3A 2K6, Canada 
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Janzen, David; Wise, Daniel T. A smallest irreducible lattice in the product of trees. Algebraic and Geometric Topology, Tome 9 (2009) no. 4, pp. 2191-2201. doi: 10.2140/agt.2009.9.2191

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