Graphs of Systoles on hyperbolic surfaces

TL;DR

This paper characterizes which fat graphs can be realized as systolic graphs on hyperbolic surfaces, establishing a necessary and sufficient combinatorial condition and showing the existence of infinitely many minimal non-admissible fat graphs.

Contribution

It introduces a combinatorial admissibility criterion for systolic graphs and proves its sufficiency, also demonstrating the infinite variety of minimal non-admissible fat graphs.

Findings

A combinatorial admissibility condition is both necessary and sufficient.

Sub-graphs of admissible graphs are also admissible.

There are infinitely many minimal non-admissible fat graphs.

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).