Graphs of systoles on hyperbolic surfaces

Abstract

Abstract. Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

Embedding of Metric Graphs on Hyperbolic Surfaces

“…The number of boundary components in Σ(G) is the number of disjoint cycles in σ 1 * σ −1 0 (see [10,Section 2.1]). For more details on fat graphs, we refer to [10,11] and [8]. 2.2.…”

“…Therefore, there is a unique geodesic representative β (simple and closed) in its free homotopy class. Note that, the geodesic representatives β of the boundary components of N (G, d, σ 0 ) are disjoint from the embedding of the graph on the surface S (see, for example, [11,Section 7]). We obtain the surface Σ 0 (G, d, σ 0 ) with totally geodesic boundary by cutting S along the geodesics in the free homotopy classes of the boundary components of N (G, d, σ 0 ).…”

“…The definition of essential graphs is motivated by the definition of essential curves on closed surfaces with negative Euler characteristic. In [11], we studied essential systolic graphs on closed hyperbolic surfaces aiming to get a natural decomposition of the moduli space of hyperbolic surfaces by associating to a surface its systolic graph.…”