Given a surface $\Sigma$ equipped with a set $P$ of marked points, we consider the triangulations of $\Sigma$ with vertex set $P$. The flip-graph of $\Sigma$ whose vertices are these triangulations, and whose edges correspond to flipping arcs appears in the study of moduli spaces and mapping class groups. We consider the number of geodesics in the flip-graph of $\Sigma$ between two triangulations as a function of their distance. We show that this number grows exponentially provided the surface has enough topology, and that in the remaining cases the growth is polynomial.

翻译：给定一个配备标记点集合 $P$ 的曲面 $\Sigma$，我们考虑以 $P$ 为顶点集的 $\Sigma$ 的三角剖分。$\Sigma$ 的翻转图出现在模空间和映射类群的研究中，其顶点对应这些三角剖分，边对应弧的翻转。我们研究 $\Sigma$ 的翻转图中两个三角剖分之间测地线的数量，并将其视为它们距离的函数。我们证明，若曲面具有足够多的拓扑结构，该数量呈指数增长；而在其余情况下，其增长为多项式形式。