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Simplicial complexity of surface groups

Published online by Cambridge University Press:  27 November 2019

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)
Corresponding

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.

Type
Research Article
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

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