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We study two 2-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed number of sides, and their geodesics diverge at most linearly.

We construct a sequence of closed hyperbolic surfaces that are local maxima for the systole function in their respective moduli spaces. Their systole is arbitrarily large and the number of examples grows rapidly with the genus. More precisely, for every n 3 there is some positive number L n (growing roughly linearly in n) such that the number of local maxima of the systole function in genus g with systole equal to L n grows super-exponentially in g along an arithmetic sequence of step size n. Many of these surfaces have no orientation-preserving isometries other than the identity and are the first examples of local maxima with this property.