  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.
