Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-29T10:17:05.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Simplicial complexity of surface groups

Part of: PL-topology Topological manifolds Global differential geometry Low-dimensional 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 ([email protected]; [email protected])
Elías Gabriel Minian
Affiliation:
Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Brown, K. S.. Cohomology of groups. Graduate Texts in Mathematics, vol. 87, pp. x+306 (New York-Berlin: Springer-Verlag, 1982).CrossRefGoogle Scholar
2Babenko, I., Balacheff, F. and Bulteau, G., Systolic geometry and simplicial complexity for groups. J. Reine. Angew. Math. (2017), published online DOI 10.1515/crelle-2017-0041.Google Scholar
3Borghini, E. and Minian, E. G.. The covering type of closed surfaces and minimal triangulations. J. Combin. Theory Ser. A 166 (2019), 1–10.Google Scholar
4Bulteau, G.. Les groupes de petite complexité simpliciale. hal-01168493. (2015).Google Scholar
5Jungerman, L. and Ringel, G.. Minimal triangulations on orientable surfaces. Acta Math. 145 (1980), 121–154.CrossRefGoogle Scholar
6Gromov, M.. Filling Riemannian manifolds. J. Diff. Geom. 18 (1983), 1–147.CrossRefGoogle Scholar
7Gromov, M.. Systoles and intersystolic inequalities. Actes de la Table Ronde de Géométrie Différentielle, Collection SMF 1 (1996), 291–362.Google Scholar
8Ratcliffe, J.. The cohomology ring of a one-relator group. In Contributions to group theory (eds. Appel, K. I., Ratcliffe, J. G. and Schupp, P. E.). Contemporary Math., vol. 33, pp. xi+519 (Providence, R.I.: Amer. Math. Soc., 1984).Google Scholar
9Ringel, G.. Wie man die geschlossenen nichtorientierbaren Flächen in möglichst wenig Dreiecke zerlegen kann. Math. Ann. 130 (1955), 317–326.CrossRefGoogle Scholar
10Rudyak, Y. and Sabourau, S.. Systolic invariants of groups and 2-complexes via Grushko decomposition. Ann. Inst. Fourier (Grenoble) 58 (2008), 777–800.CrossRefGoogle Scholar