# Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016 pp. 659–674.

Automorphisms of Non-Cyclic *p*-Gonal Riemann Surfaces

**Authors**:
Antonio F. Costa (1) and Ruben A. Hidalgo (2)

**Author institution:**(1) Departamento de Matemáticas Fundamentales, Facultad de Ciencias, UNED, 28040 Madrid, Spain

(2) Departamento de Matemática y Estadística, Universidad de La Frontera, Casilla 54-D, 4780000 Temuco, Chile

**Summary: **

In this paper we prove that the order of a holomorphic automorphism of
a non-cyclic *p*-gonal compact Riemann surface *S* of genus
*g*>(*p*−1)^{2} is bounded above by 2(*g*+*p*−1). We also show that this
maximal order is attained for infinitely many genera. This generalises
the similar result for the particular case *p*=3 recently obtained by
Costa-Izquierdo. Moreover, we also observe that the full group of
holomorphic automorphisms of *S* is either the trivial group or is a
finite cyclic group or a dihedral group or one of the Platonic groups
𝒜_{4}, 𝒜_{5} and Σ_{4}. Examples in
each case are also provided. If *S* admits a
holomorphic automorphism of order 2(*g*+*p*−1), then its full group of
automorphisms is the cyclic group generated by it and every *p*-gonal
map of *S* is necessarily simply branched.

Finally, we note that each pair (*S*,π), where *S* is a non-cyclic
*p*-gonal Riemann surface and π is a *p*-gonal map, can be defined
over its field of moduli. Also, if the group of automorphisms of *S*
is different from a non-trivial cyclic group and *g*>(*p*−1)^{2}, then
*S* can be also be defined over its field of moduli.

2010 Math. Subj. Class. 30F10; 14H37.

**Keywords:**Riemann surface, Fuchsian group, automorphisms.

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