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A characterization of Fuchsian groups acting on complex hyperbolic spaces

Xi Fu, Liulan Li, Xiantao Wang (2012)

Czechoslovak Mathematical Journal

Let G ⊂ 𝐒𝐔 ( 2 , 1 ) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ℂ -Fuchsian; if G preserves a Lagrangian plane, then G is ℝ -Fuchsian; G is Fuchsian if G is either ℂ -Fuchsian or ℝ -Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application...

An application of metric diophantine approximation in hyperbolic space to quadratic forms.

Sanju L. Velani (1994)

Publicacions Matemàtiques

For any real τ, a lim sup set WG,y(τ) of τ-(well)-approximable points is defined for discrete groups G acting on the Poincaré model of hyperbolic space. Here y is a 'distinguished point' on the sphere at infinity whose orbit under G corresponds to the rationals (which can be regarded as the orbit of the point at infinity under the modular group) in the classical theory of diophantine approximation.In this paper the Hausdorff dimension of the set WG,y(τ) is determined for geometrically finite groups...

Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps

Martine Babillot, Marc Peigné (2006)

Bulletin de la Société Mathématique de France

We consider a large class of non compact hyperbolic manifolds M = ℍ n / Γ with cusps and we prove that the winding process ( Y t ) generated by a closed 1 -form supported on a neighborhood of a cusp 𝒞 , satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp 𝒞 and the Poincaré exponent δ of Γ . No assumption on the value of δ is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez.

Big groups of automorphisms of some Klein surfaces.

Beata Mockiewicz (2002)

RACSAM

Sea Xp una superficie de Klein compacta con borde de gen algebraico p ≥ 2. Se sabe que si G es un grupo de automorfismos de Xp entonces |G| ≤ 12(p- 1). Se dice que G es un grupo grande de gen p si |G| > 4(p -1). En el presente artículo se halla una familia de enteros p para los que el único grupo grande de gen p son los grupos diédricos. Esto significa que, en términos del gen real introducido por C. L. May, para tales valores de p no existen grupos grandes de gen real p.

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