Rigidity of the saddle connection complex

Abstract

For a half-translation surface (𝑆, 𝑞), the associated saddle connection complex 𝒜(𝑆, 𝑞) is the simplicial complex where vertices are the saddle connections on (𝑆, 𝑞), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism 𝜑 : 𝒜(𝑆, 𝑞) → 𝒜(𝑆 ′ , 𝑞 ′ ) between saddle connection complexes is induced by an affine diffeomorphism 𝐹 : (𝑆, 𝑞) → (𝑆 ′ , 𝑞 ′ ). In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several purely combinatorial criteria for detecting various geometric objects on a half-translation surface, which may be of independent interest.

Large-scale geometry of the saddle connection graph

“…Recall that M q (θ) is the measured foliation on (S, q) by straight lines of slope θ ∈ RP 1 , with the measure given locally by Euclidean distance between leaves. Lemma 4.7 (Saddle connections with convergent slopes [DRT18]). Let α n ∈ A(S, q) be a sequence of saddle connections whose slopes θ n converge to θ ∈ RP 1 .…”

“…The large-scale geometry of combinatorial complexes associated to topological surfaces such as the curve graph and arc graph has played a substantial role in answering important questions on mapping class groups, Teichmüller theory, and hyperbolic 3-manifolds [BBF15, MM99, BCM12, Mos95]. More recently, analogous complexes associated to Euclidean cone metrics have attracted considerable attention; see [MT17,Ngu17,Ngu18,DRT18,Pan20,FL19]. In this paper, we consider the large-scale geometric features of the saddle connection graph A(S, q) associated to a half-translation surface (S, q); this has the saddle connections on (S, q) as vertices, with edges representing pairs of non-crossing saddle connections.…”

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]).…”