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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 -Fuchsian; if G preserves a Lagrangian plane, then G is -Fuchsian; G is Fuchsian if G is either -Fuchsian or -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...

A holomorphic correspondence at the boundary of the Klein combination locus

Shaun Bullett, Andrew Curtis (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

We investigate an explicit holomorphic correspondence on the Riemann sphere with striking dynamical behaviour: the limit set is a fractal resembling the one-skeleton of a tetrahedron and on each component of the complement of this set the correspondence behaves like a Fuchsian group.

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...

An example of degeneration on the noded Schottky space.

Rubén A. Hidalgo (1998)

Revista Matemática Complutense

In these notes we construct explicit examples of degenerations on the noded Schottky space of genus g ≥ 3. The particularity of these degenerations is the invariance under the action of a dihedral group of order 2g. More precisely, we find a two-dimensional complex manifold in the Schottky space such that all groups (including the limit ones in the noded Schottky space) admit a fixed topological action of a dihedral group of order 2g as conformal automorphisms.

Cannon-Thurston Maps, i-bounded Geometry and a Theorem of McMullen

Mahan Mj (2009/2010)

Séminaire de théorie spectrale et géométrie

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map.

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