Simplicial complexity of surface groups and systolic area

TL;DR

This paper computes the simplicial complexity of all surface groups, confirming a conjecture and providing insights into its stability under free products, thereby advancing understanding of the invariant's behavior.

Contribution

It explicitly calculates the simplicial complexity for all surface groups and demonstrates its stability under free product with free groups, addressing open problems.

Findings

Computed simplicial complexity for all surface groups.

Proved stability of simplicial complexity under free product with free groups.

Provided evidence supporting the conjecture on stability of simplicial complexity.

Abstract

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $\kappa(G)$ was proved to be a good approximation of the systolic area $\sigma(G)$ for large values of $\kappa(G)$. In this paper we compute the simplicial complexity of all surface groups (both in the orientable and in the non-orientable case). This settles a problem raised by Babenko, Balacheff and Bulteau. We also prove that $\kappa(G\ast \mathbb{Z})=\kappa(G)$ 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.