Topological Veech dichotomy and tessellations of the hyperbolic plane

Abstract

We construct for every half-translation surface satisfying the topological Veech dichotomy a tessellation of the Poincaré upper half plane. This construction can be viewed as a generalization of the Farey tessellation for a flat torus. As a consequence, we get a bound on the volume of the corresponding Teichmüller curve for a lattice surface (Veech surface). There is a natural graph underlying this tessellation on which the affine group acts by automorphisms. We provide algorithms to determine a "coarse" fundamental domain and a generating set for the Veech group based on this graph. We also show that this graph has infinite diameter and is Gromov hyperbolic.

Affine equivalence and saddle connection graphs of half-translation surfaces

“…The proof above follows essentially the argument of [4, Lemma 22]. Also, Nguyen proved similar results in [30,31].…”

“…The dual of the spine graph is a graph whose vertices are the directions of saddle connections and whose edges are pairs of directions which are the directions of the shortest saddle connections of some translation surface in the Teichmüller disk. Nguyen studied the graph of degenerate cylinders for translation surfaces in genus two ( [30]) and the graph of periodic directions for translation surface satisfying the Veech dichotomy ( [31]). He proved that both of them are hyperbolic, and that every automorphism which comes from the mapping class group is induced by an affine self-homeomorphism.…”

“…Step three is a standard argument (see [4,30,31]). The technical part of this paper is step one and step two.…”

Large-scale geometry of the saddle connection graph

“…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.…”

Rigidity of the saddle connection complex

“…Nguyen defines a graph of (degenerate) cylinders for genus 2 translation surfaces as a subgraph of the curve complex, and proves that any mapping class that stabilises it must have an affine representative [Ngu15]. He also defines a graph of graph of periodic directions in [Ngu18], and uses this to give an algorithm to compute a coarse fundamental domain for the associated Veech group. Minsky and Taylor consider a subcomplex of the arc complex defined using Veering triangulations on a half-translation surface in [MT17], and relate its geometry to that of an associated fibred hyperbolic 3-manifold.…”