The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 7

Generalized quasi-dihedral group as automorphism group of Riemann surfaces

“…Case H = DC 4n . By the proof of Theorem 4.1, the quotient orbifold S /G n has signature (0; +; [4]; {(2n)}) for n even and (0; +; [4]; {(n)}) for n odd. So,…”

“…|∆ + | * 0 = 16n(0 − 1) + 16n + 12n = 12n, from which S has genus equal to 6n + 1. Moreover, the signature (1; −; [2,2,4]; {−}) is admisible for the quotient orbifold S / G n . We proceed to check that this signature has the least reduced area.…”

“…In case ii), ∆ has signature (2; −; [4]; {−}) and an epimorphism θ : ∆ → G n is given by θ(d j ) = y t j z 2s j −1 ; θ(b 1 ) = yz 4r+2 . Then the relation θ(b 1 )θ(d 2 ) 2 θ(d 1 ) 2 = 1 implies that 1 = (yz 4r+2 )(y t 1 z 2s 1 −1 ) 2 (y t 2 z 2s 2 −1 ) 2 = yz m (a contradiction).…”

“…As C ≤ 4n, this implies that l = 0 or 1. If l = 0, then (by results in [10]), the action of the group G n with signature (1; −; [2,4]; {−}) on S can be extends to the action of a larger group on S with signature (0; +; [2]; {(2, 4)}), and again S admits reflections, and so cannot be pseudo-real. For l = 1, the signature (1; −; [2,2,4]; {−}) is the desired signature.…”

Groups of symmetric crosscap number less than or equal to 17

“…Another feature of this study is to obtain the groups which are the full automorphism group of a surface of a given genus. This was already done for g ≤ 5 in [8], for g = 6 in [4] and for g = 7 in [5].…”