Full automorphism groups of large order of compact bordered Klein surfaces

Abstract

Let S be a compact bordered Klein surface of algebraic genus g≥2, and Aut(S) its full group of automorphisms, which is known to have order at most 12(g−1). In this paper we consider groups G of automorphisms of order at least 4(g−1) acting on such surfaces, and study whether G is the full group Aut(S) or, on the contrary, the action of G extends to a larger group. The extendability of the action depends first on the NEC signature with which G acts and, in some cases, also on whether a monodromy presentation of G admits or not a particular automorphism. For each signature we study which of the three possibilities [Aut(S):G]=1, 2 or 3 occur, and show that, whenever a possibility occurs, it occurs for infinitely many values of g. We find infinite families of groups G, explicitly described by generators and relations, which satisfy the corresponding equality.