On the problem of finding the full automorphism group of a compact Klein surface

Abstract

The paper under review surveys most known results about the following problem: let $X$ be a compact topological surface of algebraic genus $p>1$, with or without boundary, orientable or not. How to calculate all groups acting as the full automorphism group of some structure of Klein surface having $X$ as underlying topological surface? It must be remarked that from Riemann's uniformization theorem, and since $\Aut(X)$ has no more than 168 $(p-1)$ automorphisms (including the orientation-reversing ones), this problem is of a finite nature. In practice this is an unaccessible task except for low values of $p$ or some extra conditions on the surfaces one is dealing with.