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.

中文翻译：

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表面组的简单复杂性

Babenko、Balacheff 和 Bulteau 最近为了研究收缩区而引入的有限可表示群的单纯复杂性是一个不变量。简单的复杂性κ(G) 被证明是收缩期的良好近似值σ(G) 对于较大的值κ(G）。在本文中，我们计算所有表面组的单纯复杂度（在可定向和不可定向情况下）。这部分解决了 Babenko、Balacheff 和 Bulteau 提出的问题。我们也证明κ(G* ℤ) =κ(G) 对于任何表面组G. 这提供了第一个部分证据，支持在具有自由基团的自由积下单纯复杂性的稳定性猜想。单纯复杂性和收缩区域的一般稳定性问题仍然悬而未决。