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GENERIC DIRICHLET POLYGONS and the MODULAR GROUP by A

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GENERIC DIRICHLET POLYGONS and the MODULAR GROUP by A GENERIC DIRICHLET POLYGONS AND THE MODULAR GROUP by A. M. MACBEATH Dedicated to Robert Rankin, teacher, colleague and friend on his seventieth birthday 1. Introduction. The concept of "marked polygon", made explicit in this paper, is implicit in all studies of the relationships between the edges and vertices of a fundamental polygon for Fuchsian group, as well as in the topology of surfaces. Once the matching of the edges under the action of the group is known, one can deduce purely combinatorially the distribution of the vertices into equivalence classes, or cycles. Knowing a little more, the order of the rotation group fixing a vertex in each cycle, we can write down a presentation for the group. It was recognized by Poincare that, when we have a polygon which is not a priori associated with a group, but when a matching of its edges is known and certain simple metric conditions, one of which he overlooked, are satisfied, then a group with the given polygon as fundamental polygon can be constructed. This is the celebrated Poincare Existence Theorem, for which a complete satisfactory proof has been published only as recently as 1971, independently by Maskit and de Rham ([6], [12]). If we choose a fixed base-point in the hyperbolic plane, then for every discrete isometry group F we can form the Dirichlet polygon by selecting only those points of the plane which are at minimal distance, among points in the same F-orbit, from p. We then have an edge-matching for this polygon. The distribution of groups in a given isomorph- ism class between the various combinatorially possible edge-matchings is of interest in itself and will probably help us to understand related questions such as the action of the Mapping Class Group on Teichmiiller space. The first obvious question to be answered is which combinatorially possible edge- matchings can actually be realized by Dirichlet polygons. In §8 we answer this question for groups with a fundamental region of finite area with a Realization Theorem which asserts that essentially all markings are possible. We use the theorem of Beardon on the existence of hyperbolic polygons with prescribed angles and having an inscribed circle. Beardon's polygon almost miraculously has precisely the properties we need to show that it is a Dirichlet polygon. In the last part of the paper we specialize to study fundamental groups of compact orientable surfaces, associated with algebraic curves. Almost all Dirichlet polygons in this case have a generic matching, that is, all vertex cycles have length 3. The technique used to prove the Realization Theorem picks out, for each generic matching, a specially symmetric Dirichlet polygon, defining what we call a central curve for each edge- matching. We prove that all central curves can be uniformized by subgroups of the modular group. 2. Notation and terminology. For the mere statement of the Realization Theorem to make sense it is essential that we should define separate entities, combinatorial and Glasgow Math. J. 27 (1985) 129-141. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.229, on 27 Sep 2021 at 17:03:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006133 130 A. M. MACBEATH geometric. The importance of this in the context of the Existence Theorem was clearly recognised by Poincare ([14], p. 127). In the context of the modular group, Millington [13] recognised that edge-matchings are essentially pairs of permutations. More recently, Hussein ([8]) made a detailed study of generic matchings in genus 2 and 3, regarding them simply as pairs of elements of Sn, n being the number of vertices of the polygon. To include orientation-reversing isometries, we here use permutations of degree In, acting on the set of directed edges. The reduction to degree n in the orientation-preserving case is made in §9. We distinguish therefore between the edge-matching, the topological marked polygon and the geometric marked polygon. We also distinguish two concepts of "polygon"— topological and geometric. The geometric polygons are always convex, for that is all we need, and only their extreme points are counted as vertices. We use the term "Dirichlet polygon" for the closure of the Dirichlet region rather than as a synonym. The closure, after all, carries all the information about the group. The term weight is used for the order of the orientable stabilizer of a vertex, rather than the word "order", which is over-used in mathematics and therefore ambiguous. Otherwise I have adhered to the terminology of [2], [9], with Magnus's notation [11] for triangle groups and Coxeter's [4], [5] for tessellations. 3. Polygons. A polygon P is a disc B2 together with a finite subset V^8B2. The points of V are the vertices of P, and, if there are n of them, P is an n-gon. We shall always assume that n^3. An edge of P is the closure of a component of 5JB2\V and a directed edge is a pair (c, u) where e is an edge and v is one of the two vertices in e. We call v the initial point or initial vertex of the directed edge. Let E(P) denote the set of In directed edges. The set of n undirected edges, when we need to refer to it, is denoted by E0(P). If (e, v) is a directed edge, there is precisely one edge e', say, other than e, which has v as a vertex, and precisely one vertex v', say, other than v, of the edge e. Define permutations p,v:E(P)^E(P) by p(e, v) = (e, v'\ v(e,v) = (e',v). The function p is direction Reversal, v(e, v) is a Neighbouring directed edge. Both p and v are regular permutations of order 2. Their composite a = vp (p first, then v) is a regular permutation of order n, a{e, v) being the Successor of (e, v) in the cyclic order of directed edges indicated by the choice of v as initial point. The set E(P), together with the pair of permutations p, v is a convenient algebraic model for the polygon. The undirected edges can be identified with elements of the quotient set E(P)l(p), the set of vertices with E(P)l(v). Working mainly with directed edges, we shall use the single Greek letter -n, with or without suffixes, for a directed edge. Then |-rj[ will denote the corresponding undirected edge and i(r\) will denote its initial point. Thus TJ = (|T)|, I'(T))). In terms of vertices, if v and v' are the two vertices of the same undirected edge e, then (v, v') will denote the directed edge (e, u). Thus (v1, v) = p(v, v'). Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.229, on 27 Sep 2021 at 17:03:56, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500006133 GENERIC DIRICHLET POLYGONS 131 An isomorphism of one n-gon (P, V) with another (P', V) is a homeomorphism of the topological pair (P, V) onto (P', V), an automorphism being as usual a self- isomorphism. Each automorphism induces a permutation of E(P) which commutes with both p and v. The group Aut(E(P)) of permutations which commute with both p and v is called the automorphism group of the set of directed edges. It is the quotient of the automorphism group of the polygon modulo those which are isotopic to the identity and can essentially be regarded as the automorphism group of P. Since (p, v) acts regularly and transitively on E(P), its action on E(P) is equivalent to the Cayley representation of the dihedral group on itself by left translation. In this model, Aut(E(P)) is the group of right translations. However, p and v are not automorph- isms of E{P) and there is no canonical isomorphism between (p, v) and the automorph- ism group. 4. Marked polygons, cycles. Let P be a polygon and let /x0 be a permutation of E0(P) such that /xo = id. Suppose that, to each e<=E0(P) there is assigned a homeomorph- ism Tie): e —» fxo(e) such that e)) = T(e)-1. (4.1) The collection (P, (x0, {T(e): e e E0(P)}) is called a topologically marked polygon and (fi,0, {T(e)}) is a topological marking of P. The homeomorphisms T(e) are called matching homeomorphisms. If v, v' are the two vertices of an edge e, then T(e) maps these to vertices vu v[ of (xo(e) respectively, thus inducing a permutation (x of directed edges given by (x(t), v') = («!, v[). Clearly ^p = pfji and, by 4.1, |x2 = id. DEFINITION 4.2. An edge-matching of P is a permutation /x :E(P)^>E(P) such that ja2 = id, jnp = p|Li. A marked polygon is a pair (P,/x) where P is a polygon and JA an edge-matching. A topologically marked polygon induces an edge-matching (x. The matching /x determines the pairing fx0 of undirected edges and we shall often write (P, /x, {T(e)}) instead of (P, /xo, {T(e)}). Unlike v, p, neither /xo nor /x need be regular permutations. In the classical case of groups generated by reflections, /x = /xo = id. DEFINITION 4.3. The quotient space of the topologically marked polygon is the space P/q where q is the equivalence relation generated by identifying T(e)x with x for each point x on each edge e.
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