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Simplicial complexity of surface groups

Part of: Topological manifolds Low-dimensional topology Global differential geometry PL-topology

Published online by Cambridge University Press:  27 November 2019

Eugenio Borghini  and
Elías Gabriel Minian
Eugenio Borghini
Affiliation:
Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (eborghini@dm.uba.ar; gminian@dm.uba.ar)
Elías Gabriel Minian
Affiliation:
Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina (eborghini@dm.uba.ar; gminian@dm.uba.ar)
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Abstract

The simplicial complexity is an invariant for finitely presentable groups which was recently introduced by Babenko, Balacheff, and Bulteau to study systolic area. The simplicial complexity κ(G) was proved to be a good approximation of the systolic area σ(G) for large values of κ(G). In this paper we compute the simplicial complexity of all surface groups (both in the orientable and in the non-orientable case). This partially settles a problem raised by Babenko, Balacheff, and Bulteau. We also prove that κ(G * ℤ) = κ(G) for any surface group G. This provides the first partial evidence in favor of the conjecture of the stability of the simplicial complexity under free product with free groups. The general stability problem, both for simplicial complexity and for systolic area, remains open.

Keywords

systolic areasimplicial complexitysurface groups

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 57M20: Two-dimensional complexes 57Q15: Triangulating manifolds 57N16: Geometric structures on manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3153 - 3162
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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