Affine mappings of translation surfaces: geometry and arithmetic

Abstract

and CHRIS JUDGE 1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teichmüller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurston's classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points.

Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

“…general flat structures are the half-translation surfaces [11]. All of our results extend mutatis mutandis to the halftranslation surfaces.…”

“…The other is the study of special translation surfaces, e.g., lattice surfaces. This branch naturally subdivides: The purely geometric one [9,29] and the algebro-geometric one [10,11,14,15,17,30]. The present work is of the latter type.…”

“…A point of S is G-periodic if its G-orbit is finite. When G = Aff(S), we simply speak of the periodic points of S. It follows from Theorem 5.5 of [11] that the periodic points of an arithmetic translation surface form a countable, dense set. THEOREM 1.…”

Bill Veech's contributions to dynamical systems

“…However, in every stratum there are countably many arithmetic lattice surfaces, that is, surfaces with isotropy group commensurable with a conjugate of SL (2, Z). By a theorem of E. Gutkin and C. Judge [40] a surface is arithmetic if and only if it is a branched cover of a torus with a single branch point (or, equivalently, if and only if it is tiled by parallelograms). Non-arithmetic examples are much harder to find.…”

Generic absence of finite blocking for interior points of Birkhoff billiards

“…According to Gutkin [11], for a regular n-gon to be secure, it is necessary and sufficient that n = 3, 4, or 6. The proof of necessity is deep and is based on earlier work on security in translation surfaces [10,25,26,14,15,13].…”