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A celebrated result of Mirzakhani states that, if $(S,m)$ is a finite area\emph{orientable} hyperbolic surface, then the number of simple closedgeodesics of length less than $L$ on $(S,m)$ is asymptotically equivalent to apositive constant times $L^{\dim\mathcal{ML}(S)}$, where $\mathcal{ML}(S)$denotes the space of measured laminations on $S$. We observed on some explicitexamples that this result does not hold for \emph{nonorientable} hyperbolicsurfaces. The aim of this article is to explain this surprising phenomenon. Let$(S,m)$ be a finite area \emph{nonorientable} hyperbolic surface. We show thatthe set of measured laminations with a closed one--sided leaf has a peculiarstructure. As a consequence, the action of the mapping class group on theprojective space of measured laminations is not minimal. We determine a partialclassification of its orbit closures, and we deduce that the number of simpleclosed geodesics of length less than $L$ on $(S,m)$ is negligible compared to$L^{\dim\mathcal{ML}(S)}$. We extend this result to general multicurves. Thenwe focus on the geometry of the moduli space. We prove that its Teichm\"ullervolume is infinite, and that the Teichm\"uller flow is not ergodic. We alsoconsider a volume form introduced by Norbury. We show that it is the rightgeneralization of the Weil--Petersson volume form. The volume of the modulispace with respect to this volume form is again infinite (as shown by Norbury),but the subset of hyperbolic surfaces whose one--sided geodesics have length atleast $\varepsilon>0$ has finite volume. These results suggest that the modulispace of a nonorientable surface looks like an infinite volume geometricallyfinite orbifold. We discuss this analogy and formulate some conjectures.

What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces