Journal of Mathematical Sciences, Vol. 82. No. 6, 1996 ON THE AUTOMORPHISM GROUP OF THE CANONICAL DOUBLE COVERING OF BORDERED KLEIN SURFACES WITH LARGE AUTOMORPHISM GROUP* A. M. Porto Fereira da Silva and A. F. Costa UDC 512.542,512.772,515.177.4,517.545,514.133 Introduction Assume that X is a Klein surface with a nonempty boundary and X + is its double covering which is a Riemannian surface. We denote by Aut X the group of automorphisms of X and by ~ut X + the group of conformal and anticonformal automorphisms of X +. We want to get information about Aut X + from Aut X. In terms of algebraic geometry, we can say that the problem consists in studying a group of automorphisms of a smooth complex algebraic curve from the knowledge of a group of automorphisms of its real part. It has been proved recently (see [4] and [2]) that if the order of Aut X is high as compared to the algebraic genus g of X, more exactly, if #Aut X > 8(g- 1), then Aut X + = Aut X x C2 (the direct product of Aut X by the cyclic group of order 2) except for a finite number of Klein surfaces. In this paper, we obtain the following result, which goes further than the previous one: if #AutX > 6(9 - 1), then AutX + = AutX • C2 except for a finite number of cases (Corollary 2.1). In Sec. 3, we prove that every irreflexible regular triangular map provides a Klein surface X such that Aut X + ~ Aut X • C2 and #Aut X = 6(g" 1). Then, as a consequence of [7], there are infinitely many Klein surfaces with different topological types under the above conditions. In addition, we show that if there is an automorphism with fixed points and prime order p of X and a--~. 4p > #Aut X, then Aut X + = Aut X • C~ but for a unique exception for every value of g. p--2 In order to establish the above results, we study in See. 1 the non-Euclidean crystallographic groups generated by reflections on the geodesics containing the sides of a hyperbolic Lambert quadrilaterial whose nonright angle is r/p, where p is an odd prime. 1. Lambert's Quadrilaterals A Lambert quadrilaterial is a hyperbolic quadrilaterial with angles rr[2, a'/2, ~r/2, and ~b (see [1]). In this paper, we study the groups generated by the reflections on the sides (i.e., on the geodesics containing the sides) of a Lambert quadrilateriai such that r = 7rip with p being an odd prime. We denote such a quadfilaterial by Cp and say that the group generated by the reflections on the lines containing the sides of Cp is a group of type (2,2,2,p). These groups are a particular case of non-Euclidean crystallographic (NEC) groups; the following lemma concerns NEC groups which contain Cp as a subgroup (see [3] for terminology and notations). Lemma 1.1. Suppose that p is a prime, p > 2, and F is a group of type (2,2,2,p) defined by the Lambert quadrilaterial ABCD of Fig. 1. If we assume that F is strictly contained in another NEC group A, then the length of the side BC of ABCD is the same as that of CD and is equal to arccosh (v~cos ~); moreover, the group A is the group generated by the reflections on the sides of the triangle ABC (Fig. 1). * Supported by DCICYT PB89-0201, Science Program CEE EttB 4002 PL 910021, and Acciones Integradas Hispano-Portuguesas. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 21, Algebra-3, 1995. 1072-3374/96/8206-3773 $15.00 9 1996 Plenum Publishing Corporation 3773