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Canonical Polygons for Finitely Generated Fuchsian Groups

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Canonical Polygons for Finitely Generated Fuchsian Groups CANONICAL POLYGONS FOR FINITELY GENERATED FUCHSIAN GROUPS BY LINDA KEEN The Institute for Advanced Study, Princeton, N.J., U. S. A. 1. Introduction We consider finitely generated Fuchsian groups G. For such groups Fricke defined a class of fundamental polygons which he called canonical. The two most important distinguishing properties of these polygons are that they are strictly convex and have the smallest possible number of sides. A canonical polygon P in Fricke's sense de- pends on a choice of a certain "standard" system of generators S for G. Fricke proved that for a given G and S canonical polygons always exist. His proof is rather com- plicated. Also, Fricke's polygons are not canonical in the technical sense; there are infinitely many P for a given G and S. In this paper, we shall construct a uniquely determined fundamental polygon P which satisfies all of Fricke's conditions for every given G of positive genus and for every given S. We call this P a canonical Fricke polygon. For every given G of genus zero, and S, we shall define a uniquely determined fundamental polygon P which we call a canonical polygon without accidental vertices. From this P, one can obtain in infinitely many ways polygons satisfying l~'ricke's conditions. Our canonical polygons are invariant under similarity transformations of the group G if G is of the first kind. If G is of the second kind, this statement remains true after a suitable modification which will be clear from the construction. The proof involves elementary explicit constructions, and continuity arguments which use quasiconformal mappings, as developed by Ahlfors and Bers. We give a geometric interpretation of the canonical polygons in the last section. This interpretation provided the heuristic idea for the formulation of the main theorem. 1 --652944 Acta mathematica. 115. Imprim6 le janvier 1966 2 LINDA KEEN The author takes pride and pleasure in acknowledging the guidance of professor Lipman Bers in the preparation of this dissertation. His patient and penetrating counsel and his warm friendship will be remembered always. 2. Definitions We say that S is a Riemann surface of finite type (g; n; m) if it is conformally equivalent to ~- ((Pl, P2 ..... Pn) U (d l, d~ ..... din)} where ~ is a closed Riemann sur- face of genus g, the p~ are points and the d s are closed conformal discs (p~:~pj, d~ N dj=O for i:~j, p~dj, n>~0, m~>0). Suppose that to each "removed" point pj, ~= 1, 2, ..., n, there is assigned an "integer" vj, rs = 2, 3, ..., co, ~1 ~<u~ ~< ... ~<r~. Then we say that S has signature (g; n; r 1..... v.; m). A surface's type is preserved under conformal equivalence, hence this definition is meaningful. A surface S with signature (g; n; ~1 ..... u~; m) is represented by a Fuchsian group G if: 1) G is a properly discontinous group of MSbius transformations leaving the unit disc U fixed. 2) If Ua denotes U-(elliptic fixed points of G} then 2a) U/G is conformally equivalent to ~ - ((p ..... , p,) U (d 1..... din)} where rr = rr+l = ... = co. Ua/G is eonformally equivalent to ~-((Pl ..... p~) U (d 1..... d~)}. 2b) The map ga:U--> U/G is locally 1 to 1 in the neighborhood of every point of Ua, and is ~s to 1 at the pre-images of the points Pl, .--,P~-I. 3. Preliminaries From now on we consider only Riemann surfaces of finite type. The following three classical theorems are basic to the theory we are discussing. T~ E ORE M 1. Given a Riemann sur/ace with a signature, a Fuchsian group representing it is finitely generated and is determined up to conjugation by a MSbius trans/ormation. The proof may be found in Appell-Goursat [4]. THEO~.M 2. All /initely generated Fuchsian groups represent sur/aces o//inite type. A direct proof will appear in a forthcoming paper by Bers [7], and can also be found as a special case of a theorem of Ahlfors [2]. CANONICAL POLYGONS FOR FINITELY GENERATED FUCHSIAN GROUPS THE ORE M 3. There exists a/initely generated Fuchsian group representing every sur- ]ace o/ /inite type with a given signature provided that 3g- 3 + n + m > 0 and i/ g = 0, m = O, n = 4 then ~=1 vj > 8. Note that we omit any discussion of the triangle groups in this paper. Proof of this may be found in Ford [9] and Bers [5]. However, the statemen~ also follows from the construction we will give later. We will sketch the proof at the appropriate place. 4. Frieke polygons If (g; n; v 1..... vn; m) is the signature of S, and if G is the group representing S, we call (g; n; Vl .... , Vn; m) the signature of G. We remark that the signature of G is preserved under conjugation. Let G be a finitely generated Fuchsian group with signature (g; n; vl .... , vn; m) and suppose R is a fundamental region for G. R is called a standard [undamental region for G if it satisfies the following conditions: 1) R is bounded by 4g + 2n + 2m Jordan arcs in U and m arcs on the boundary of U, forming a Jordan curve oriented so that the interior of R is on the left. 2) If the sides of R are suitably labelled in order: I al, 51, al,' b"1~ as, b~, a~, b~, ..., ag' bg," cl, c~, ..., On, en," dl, el, d;, ..., d,n, era, d,n, there exist hyperbolic elements As, B~ and DjEG, i= 1, 2 ..... g, j= l, 2 ..... m, such that A~(a~)= -a'~, B~(b;)= -b~ and Dj(dj)= -d;, and elliptic elements CkeG of order vk (parabolic if vk = oo) k = 1, 2 ..... n, such that Ck(c~)= -c'k. These elements satisfy the relation: D~ ... DI C~ ... C1B~I A~I BgA~ ... B11A11Bs A1 = 1. (1) (Note: In this paper the notation AB means first apply B and then apply A.) The ej, j= 1, 2 ..... ra are arcs on the unit circle. It is known that the elements of S = {A 1, B 1 .... Ao, Bo, C~ .... Cn, Dl ..... D,n} with the single relation (I), generate G. ~re will call S a standard sequence o/genera- tors. R is said to belong to S. Two standard fundamental regions are equivalent if they give rise to the same standard sequence of generators. 4 LINDA KEEN It follows easily from Theorem 2 that standard fundamental regions exist for all finitely generated Fuchsian groups. However, this will also follow from the main theorem of this paper; an explicit proof using this theorem will be given later (see Section 7). If the boundary arcs of a standard fundamental region R belonging to $ (except for the arcs on the unit circle) are non-Euclidean straight segments, R will be called a Fricke polygon belonging to S. 5. Fricke's theorem Let $ be a standard sequence of generators and let P0 be a point in U. We will construct a closed "polygonal" curve P(P0; ~) such that if this curve has no self-intersections, it is the boundary of a Fricke polygon belonging to $. Let Po be given. Find the points: P~ = Af *Bf ~(P0), P~ = Af ~(P0), Pg = Bf 1(P0), P~ = Pl = Bf l A f l (Po) Pl = A21B~A~(p~), .p~ = BeA~(pl), p~ = A~(pl), P~ =P2 = B~I A~I B~A~(pl) P~=A~IBaAa(p~), p~=BaAz(p~), P~= A a(P~), P~ =Pa = B31Aa1BaAa(Pu), P~-I = A~IBg Ag (Pg-1), P2a-1 = Bg Ag (Pg-1), P~-~ = Ao (Pg-1), P~-I = Pg = B~A~ 1Bg Ag(pg_~) Pg+I = Cl (p~), P~+2 = C2(p~+~) ..... p~+, = C, (p~+~-l), Pg+~+l=Dl(pg+,), pg+~+~=D2(pg+~+l) ..... pa+,+m-po-Dm(Pg+,+m), and label the fixed points of C1, C~ ..... C, by ql, q2 ..... q~. If Ij is the isometric circle (see Ford [9]) of Dj, Ij intersects the axis of Dj. This axis has a direction in which it is moved by the translation D s. Let r~ be the endpoint of Ij on the left side of the axis of Ds. Then r~=Dj(rj) is also to the left of the axis of D s. Join by non-Euclidean straight segments (see Figure 1): pl to p~, p~ to p0, po to p~, p~ to p~ =pl pl to pl, pl to p~, p~ to p~, p~ to pl =p~ P2 to p~, ... .. P~-I to P$-1 =Pg pg to ql, ql topo+l, Pa+l to q2 ..... qn topg+n Pg+n to rl, r~ to Pg+.+l, Po+n+l to r2, r~ to Pg+n+u..... r~ to Po+n+m =pl. CANONICAL POLYGONS FOR FINITELY GENERATED FUCHSIAN GROUPS e2t R / q2 0 p Fig. 1 Fig. 2 Fig. 1. R is a standard fundamental region belonging to S = {Ax, B1, A,, B2, e l, C2, D x, D2} where S generates the group G with signature (2; 2; k, oo; 2). R is in fact a Fricke polygon. Fig. 2. G has signature (2; 2; k, co ; 0). We call the resulting polygonal curve P(190; S). If it bounds a fundamental re- gion for G (and hence is a Fricke polygon) we call 190 suitable. If P(19o; $) is also strictly convex (i. e. all the interior angles are strictly less than 7e, except for those at vertices which are fixed points of elliptic transformations of order 2) we call P0 very suitable.
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