Affine equivalence and saddle connection graphs of half-translation surfaces

Abstract

To every translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying translation surfaces. We also investigate the automorphism group of the saddle connection graph, and the corresponding quotient graph.

Rigidity of the saddle connection complex

“…During the final writing stages of this paper, Huiping Pan gave an independent proof of Theorem 1 in the case of translation surfaces [Pan18]. The overall strategy of Pan's proof is similar to ours.…”

“…By following a standard method of Luo (which is adapted from an argument of Kobayashi [Kob88] and appears in [MM99]), Pan shows that 𝒜(𝑆, 𝑞) has infinite diameter [Pan18]. A consequence of Proposition 3.4 is that i 𝑞 induces an embedding of the Gromov boundary of 𝒜(𝑆, 𝑞) into that of 𝒜(𝑆, 𝒵).…”

Cylinder curves in finite holonomy flat metrics

“…For such metrics, the set of cylinder curves is an important tool in identifying the metric up to affine deformation, leading to rigidity results of the second author with Duchin and Rafi [6] and Loving [12]. For half-translation surfaces, combinatorial and geometric properties of cylinder curves and saddle connections, viewed as subsets of the curve graph and arc graph have recently been recently studied by Tang-Webb [20], Pan [15] and Disarlo-Randecker-Tang [5], as well as forthcoming work of Tang [19]. The connection between flat metrics and the curve graph has its origin in the work of Masur and Minsky [14] (see also Bowditch [3]), while more generally, cylinders in flat metrics arose naturally in complex analysis via extremal problems (see Strebel [18]) and in dynamics of rational billiards via periodic billiard trajectories (see, for example, Masur [13] and Boshernitzan-Galperin-Krüger-Troubetzkoy [2]).…”