Riemann surfaces with real forms which have maximal cyclic symmetry

Abstract

We determine all the Klein surfaces which have a cyclic automorphism group of the maximum possible order, and find their topological types. We also compute their full automorphism groups and show that, except for a finite number of exceptions, they coincide with the full automorphism groups of their Riemann double covers. Explicit algebraic equations of the surfaces and the formulae of their real forms and automorphisms are also given.  2004 Elsevier Inc. All rights reserved.

Extensions of finite cyclic group actions on bordered surfaces

“…It is also known that such a maximal cyclic action on a bordered surface always extends to a larger group of automorphisms; see [5]. We can now give a direct proof of this result, as a corollary of Theorem 2.1.…”

“…We can now give a direct proof of this result, as a corollary of Theorem 2.1. It is worth mentioning that the full automorphism group of such a surface is a dihedral 2-extension of v , except for three surfaces of odd genus and one of even genus; see [5].…”

“…. ., 2)}) when g = 2 +3 or (0; +; [3,3]; {(2, 2)}) when g = 6 +3 or (0; +; [2,3,3]; {(−)}) when g = 6 +3 or +1 or (0; +; [2,3,4]; {(−)}) when g = 12 +1 or (0; +; [3,4]; {(2, 2)}) when g = 12 −6 or (0; +; [3,5]; {(2, 2)}) when g = 30 +15 or (0; +; [2,3,5];…”

“…In either case, θ(e 1 ) = θ(d 1 ) −2 has order n/2, and so σ(ker θ) has two empty period cycles. With regard to orientability, ker θ always contains a non-orientable word, such as d (3,4), (3,5), (3,6) and (4,4). In all cases, n = lcm(m 1 , m 2 , 2), since C n is generated by elements of orders m 1 , m 2 and 2; for example, in the first case m = n or m = n/2, with the latter value occurring only when n/2 is odd.…”

“…(0; +; [3,3]; {(2, 2)}), n= g = 6; (0; +; [2, n/2], {(2, 2)}), n= g + 1≡ 2 (mod 4); (0; +; [3,5]; {(2, 2)}), n= g = 30; (0; +; [3,4]; {(2, 2)}), n = g = 12; (0; +; [3,6];…”