Compact Klein surfaces of genus 5 with a unique extremal disc

Abstract

A compact (orientable or non-orientable) surface of genus g g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g g . The present paper concerns non-orientable extremal surfaces of genus 5 5 . We represent the surfaces as side-pairing patterns of a hyperbolic regular 24 24 -gon, that is, a generic fundamental region of an NEC group uniformizing each of the surfaces. We also describe the group of automorphisms of the surfaces with a unique extremal disc.