The Tits Alternative for Cat(0) Cubical Complexes

By Michah Sageev, Daniel T. Wise. 2004

81 citations recorded by the source corpus.

A Tits alternative theorem is proved in this paper for groups acting on CAT(0) cubical complexes. That is, a proof is given to show that if G is assumed to be a group for which there is a bound on the orders of its finite subgroups, and if G acts properly on a finite‐dimensional CAT(0) cubical complex, then either G contains a free subgroup of rank 2, or G is finitely generated and virtually abelian. In particular, the above conclusion holds for any group G with a free action on a finite‐dimensional CAT(0) cubical complex. 2000 Mathematics Subject Classification 20F67, 20E08.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3f9569b17fdf6fca71b49a7b3a65ae86617fd3fc

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In Geometry and Topology

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Citations in the corpus

Rank Rigidity for Cat(0) Cube Complexes (2010, methodology): “This is a slight improvement of the statement in [ SW ].”
An introduction to right-angled Artin groups (2006, unknown): “For example, G is biautomatic [71] and satisfies a Tit’s alternative: any subgroup of G is either virtually abelian or contains a non-abelien free group [76].”
Finiteness properties of cubulated groups (2012, background): “If G has bounded torsion and acts properly on a finite dimensional CAT(0) cube complex then G satisfies a strong form of the Tits Alternative by Sageev– Wise [SW05].”
Boundaries and JSJ Decompositions of CAT(0)-Groups (2007, methodology): “On the other hand if X is a cube complex then the Tits alternative holds as shown by Sageev and Wise [ SW ].”