Maximal and Non-maximal NEC and Fuchsian groups uniformizing Klein and Riemann surfaces

Topologically singular points in the moduli space of Riemann surfaces

“…Proof Note that every stratum of B g is contained in the closure S k of some equisymmetric stratum defined by the action of a prime order automorphism (for instance see [3], [4], [5]). We shall then study the dimension of such strata.…”

“…The stratum S has a neighbourhood that is homeomorphic to the quotient space of a ball by the action of an order two rotation with fixed point set of codimension 2, then the points in S has dimension 6 × 1 − 6 + 2 × 4 = 8 (codimension > 2). Or the signature (1; 2, 2, 2, 2) is maximal (see for instance [5]), then this stratum is maximal. By Corollary 2 all the points in S…”

“…Type C: for this type of automorphisms there are no fixed points and the quotient surface has genus 2. This stratum is not maximal since all Riemann surfaces of genus three with a fixed point free involution are hyperelliptic and as well admit an involution with four fixed points (the signature of Fuchsian groups determining this stratum is not maximal, see [5]). Hence S and all its points are singular.…”