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We prove a rigidity theorem for semiarithmetic Fuchsian groups: If Γ₁, Γ₂ are two semiarithmetic lattices in PSL(2,ℝ ) virtually admitting modular embeddings, and f: Γ₁ → Γ₂ is a group isomorphism that respects the notion of congruence subgroups, then f is induced by an inner automorphism of PGL(2,ℝ ).

Modular embeddings for some non-arithmetic Fuchsian groups

Paula Cohen, Jürgen Wolfart (1990)

Acta Arithmetica

Noncongruence subgroups of the Hecke groups G (...).

Gareth A. Jones (1980)

Journal für die reine und angewandte Mathematik

Non-maximal cyclic group actions on compact Riemann surfaces.

David Singerman, Paul Watson (1997)

Revista Matemática de la Universidad Complutense de Madrid

We say that a finite group G of automorphisms of a Riemann surface X is non-maximal in genus g if (i) G acts as a group of automorphisms of some compact Riemann surface Xg of genus g and (ii), for all such surfaces Xg , |Aut Xg| > |G|. In this paper we investigate the case where G is a cyclic group Cn of order n. If Cn acts on only finitely many surfaces of genus g, then we completely solve the problem of finding all such pairs (n,g).

On a Conjecture of Magnus on the Hurwitz Monodromy Group.

Colin Maclachlan (1973)

Mathematische Zeitschrift

On arithmetic Fuchsian groups and their characterizations

Slavyana Geninska (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

This is a small survey paper about connections between the arithmetic and geometric properties in the case of arithmetic Fuchsian groups.

On compact Klein surfaces with a special automorphism group.

Bujalance, E. (1997)

Annales Academiae Scientiarum Fennicae. Mathematica

On Cyclic Groups of Möbius Transformations.

Troels Jorgensen (1973)

Mathematica Scandinavica

On fixed points of automorphisms of non-orientable unbordered Klein surfaces.

G. Gromadzki (2009)

Publicacions Matemàtiques

On geometric convergence of discrete groups

Shihai Yang (2014)

Czechoslovak Mathematical Journal

One of the basic questions in the Kleinian group theory is to understand both algebraic and geometric limiting behavior of sequences of discrete subgroups. In this paper we consider the geometric convergence in the setting of the isometric group of the real or complex hyperbolic space. It is known that if $Γ$ is a non-elementary finitely generated group and $ρ_{i} : Γ \to SO (n, 1)$ a sequence of discrete and faithful representations, then the geometric limit of $ρ_{i} (Γ)$ is a discrete subgroup of $SO (n, 1)$ . We generalize this result by...

On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number.

Leininger, Christopher J. (2004)

Geometry & Topology

On Klein Surfaces and Dihedral Groups.

M. Izquierdo (1995)

Mathematica Scandinavica

On limit sets of geometrically finite Kleinian groups.

Pekka Tukia (1985)

Mathematica Scandinavica

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

Ewa Tyszkowska (2003)

Open Mathematics

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p n are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain...

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