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On the girth of groups
acting on CAT(0) cube complexes
Arka Banerjee, Daniel L Gulbrandsen, Pratyush Mishra and Prayagdeep Parija |
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Algebraic & Geometric Topology 26 (2026) 2539–2558
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Abstract |
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We obtain a sufficient condition for lattices in the automorphism group of a finite-dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of the girth alternative for groups acting geometrically on CAT(0) cube complexes: any such group is either {locally finite}-by-{virtually abelian} or it has infinite girth. We produce counterexamples to show that the alternative fails in the general class of groups acting cocompactly on finite-dimensional CAT(0) cube complexes using examples of nonvirtually solvable groups which satisfy a law. |
Keywords
girth alternative, CAT(0) cube complex
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Mathematical Subject Classification
Primary: 20F65
Secondary: 20F67
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References |
Publication
Received: 25 August 2024
Revised: 27 May 2025
Accepted: 27 July 2025
Published: 1 September 2026
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Authors |
| Arka Banerjee | |
| Department of Mathematics and
Statistics Auburn University Auburn, AL United States |
Department of Mathematics Ramakrishna Mission Vivekananda Educational and Research Institute Belur, West Bengal India |
| Daniel L Gulbrandsen | |
| Department of Mathematics Utah Valley University Orem, UT United States |
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| Pratyush Mishra | |
| Department of Mathematics Wake Forest University Winston-Salem, NC United States |
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| Prayagdeep Parija | |
| Department of Mathematics Virginia Tech Blacksburg, VA United States |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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