Bordered Klein surfaces with maximal symmetry

Abstract

Abstract. A compact bordered Klein surface of (algebraic) genus g > 2 is said to have maximal symmetry if its automorphism group is of order 12(g -1), the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the A/*-groups. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated M*-group. We begin by classifying the bordered surfaces with maximal symmetry of low topological genus. We next show that a bordered surface with maximal symmetry is a full covering of another surface with primitive maximal symmetry. A surface has primitive maximal symmetry if its automorphism group is M*-simple, that is, if its automorphism group has no proper M*-quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maxima] symmetry. Next we obtain several constructions of full covers of a bordered surface. These constructions give numerous infinite families of surfaces with maximal symmetry. We also prove that only two of the M*-simple groups are solvable, and we exhibit infinitely many nonsolvable ones. Finally we show that there is a correspondence between bordered Klein surfaces with maximal symmetry and regular triangulations of surfaces. 0. Introduction. In the fundamental paper [6] Hurwitz showed that a compact Riemann surface of genus g > 2 has at most 84(g -1) automorphisms. Recent research [7,8,16] has studied the values of g for which this bound is attained and the structure of the automorphism group in these cases.A compact bordered Klein surface [1] of genus g>2 has at most 12(g-1) automorphisms [10]. In this paper we study the surfaces for which this bound is attained, the bordered surfaces which have "maximal symmetry." We also examine the automorphism groups of these surfaces, the M*-groups [11]. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated Af*-group. §1 contains preliminary results and definitions, while §2 classifies the bordered surfaces with maximal symmetry of low topological genus.§3 is the central section of the paper. Surfaces with maximal symmetry are seen to be "full" coverings of those with "primitive" maximal symmetry. The latter surfaces are those whose automorphism groups are "A/*-simple." The classification problem for surfaces with maximal symmetry then breaks into three parts, which are considered in the next three sections.

Complex doubles of bordered Klein surfaces with maximal symmetry

“…There is a strong connection here with the theory of regular maps on surfaces. We are using "regular" in the usual way [4], not in the strong sense of [6]. Large groups of automorphisms of Riemann surfaces correspond to groups of regular maps.…”

“…But the order of UV is equal to the order of 6(UV) = JT, which is two. Hence q = 2 and H = D 6 , the only M*-group with index 2 [9, p. 377]. With this index, D 6 acts on a unique topological type of bordered surface, a sphere with three holes.…”

“…There must be Riemann surfaces of genus two with automorphism group C 2 x D 6 that are complex doubles of orientable surfaces with three boundary components, but no surface with species +3 appears in the table [3, p. 518]. However, Singerman has informed us [16] that the table should have an additional entry, a surface X with A + (X) = D 6 , A(X) = C 2 xD 6 , UCT group (0;+; [2,2,2,3]; { }), EUCT group (0; +; [ ]; {(2,2,2,3)}) and symmetry type {0, +1, +3,+3}.…”

“…A compact bordered Klein surface X of algebraic genus g > 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g -1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ( [6] and [9] among others).…”

“…The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ( [6] and [9] among others).…”