Automorphism groups of Klein surfaces without involutions

Abstract

The authors describe in terms of non-Euclidean crystallographic groups all Klein surfaces whose automorphism group is one of the following: Z/p⊕⋯⊕Z/p , Z/pq , or Z/p 2 , where p and q are distinct odd primes. This includes every nontrivial finite group of order less than 21, so they are able to use their results to find all topological types of Klein surfaces of algebraic genus less than 22 whose automorphism group has odd order bigger than one. This list takes 29 pages! They note that the cyclic groups of orders 13, 17 and 19 do not appear, a result of some interest as these groups certainly act as a subgroup of the automorphism group of a surface of algebraic genus less than 22.