Full automorphism groups of large order of compact bordered Klein surfaces

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. (c) 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).