On Riemann surfaces of genus g with 4g automorphisms

Abstract

We determine, for all genus g ≥ 2 the Riemann surfaces of genus g with 4g automorphisms. For g = 3, 6, 12, 15 or 30, this surfaces form a real Riemann surface F g in the moduli space M g : the Riemann sphere with three punctures. The set of real Riemann surfaces in F g consists of three intervals its closure in the Deligne-Mumford compactification of M g is a closed Jordan curve.

A note on large automorphism groups of compact Riemann surfaces

“…cannot be deformed non-trivially in the moduli space together with its automorphisms) or belong to a complex one-dimensional family. See [7,11,12,24].…”

One-dimensional families of Riemann surfaces of genus g with $$4\text {g}+4$$ 4 g + 4 automorphims

“…Following the technics in [6] and [7] and studying the surfaces admitting real forms in the family, we may announce the following result:…”

“…For surfaces of genus g with automorphism group of order 4g, if g is greater than 30, all of them are in an equisymmetric uniparametric family (see [7]). This phenomena does not happen for surfaces with 4g + 4 automorphism.…”

“…We prove that the maximum number ag + b of automorphisms of generic Riemann surfaces in equisymmetric and (complex)-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4 (see Theorem 1). The second possible largest number of automorphisms where this fact is verified is 4g and this case is studied in [7].…”

“…For surfaces of genus g with automorphism group of order 4g and g > 30 the automorphism group is isomorphic to D 2g ( [7]). For families of surfaces of genus g with 4g + 4 automorphisms there are surfaces of infinitely many genera with automorphism group non-isomorphic to the automorphism group of the family A g .…”