2019/02/08 by Rubén A. Hidalgo, Hidalgo, Ruben A.
Mathematics · #14H37 #30F10 #30F40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1902.03286
openalex publication_date 2019/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A group H ≅ \mathbb Zk2g, where g,k ≥ 2 are integers, of conformal automorphisms of a closed Riemann surface S is called a (g,k)-Fermat group if it acts freely with quotient S/H of genus g. We study some properties of these type of objects, in particular, we observe that S is non-hyperelliptic and, if k=pr, where p>84(g-1) is a prime integer and r ≥ 1, then H is the unique (g,k)-Fermat group of S. Let Γ be a co-compact torsion free Fuchsian group such that S/H=\mathbb H2/Γ. If Γk is its normal subgroup generated by its commutators and the k-powers of its elements, then there is a biholomorphism between S and \mathbb H2/Γk congugating H to Γ/Γk. The inclusion Γk < Γ induces a natural holomorphic embedding Θk:\mathcal T(Γ) \hookrightarrow \mathcal T(Γk) of the corresponding Teichmüller spaces. Such an embedding induces a holomorphic map, at the level of their moduli spaces, Φk:\mathcal M(Γ) → \mathcal M(Γk). As a consequence of the results on (g,k)-Fermat groups, we provide sufficient conditions for the injectivity of Φk.