The complexity of a branched cover of a Riemann surface M to the Riemann sphere S2 is defined as its degree times the hyperbolic area of the complement of its branching set in S2. The complexity of M is defined as the infimum of the complexities of all branched covers of M to S2. We prove that if M is a connected, closed, orientable Riemann surface of genus g≥1, then its complexity equals 2π(mmin+2g−2) , where mmin is the minimum total length of a branch datum realisable by a branched cover p:M→S2.

Let G⊂SU(2,1) be nonelementary and S be its minimal generating system. In this paper, we show that if S satisfies some conditions, then S can be replaced by a minimal generating system S1consisting only of loxodromic elements.

Let {Gr,i} be a sequence of r-generator Kleinian groups acting on . In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G1 and G2 of an infinite co-volume Kleinian group G⊂Isom(H3) having Λ(G1)=Λ(G2) are commensurable. In particular, we prove that any finitely generated subgroup H of a Kleinian group G⊂Isom(H3) with Λ(H)=Λ(G) is of finite index if and only if H is not a virtually fibered subgroup.

In this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

Let S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

There are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

J. W. Anderson (1996) asked whether two finitely generated Kleinian groups with the same set of axes are commensurable. We give some partial solutions.

In this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

In this paper, one model of the universal Teichmüller space is studied. By the method of construction, the lower bound of the inner radius of univalency by the Pre-Schwarzian derivative of quasidisks with infinity as an inner point (such as domains bounded by ellipses) is obtained.

We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Möbius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the d–sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.

In this paper, we shall show that the constant in Smale's mean value theorem can be reduced to two for a large class of polynomials which includes the odd polynomials with nonzero linear term.

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3 with Markoff triples.

The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

Two projective nonsingular complex algebraic curves X and Y defined over the field R of real numbers can be isomorphic while their sets X(R) and Y(R) of R-rational points could be even non homeomorphic. This leads to the count of the number of real forms of a complex algebraic curve X, that is, those nonisomorphic real algebraic curves whose complexifications are isomorphic to X. In this paper we compute, as a function of genus, the maximum number of such real forms that a complex algebraic curve admits.

We outline the classification, up to isometry, of all tetrahedra in hyperbolic space with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncations are all π/2, and those remaining are all submultiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups.

For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary. In particular, for each g ≥ 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such manifolds.

The parabolicity of Brelot's harmonic spaces is characterized by the fact that every positive harmonic function is of minimal growth at the ideal boundary.

Deformation spaces of quasi-Fuchsian groups provide the simplest nontrivial examples of deformation spaces of Kleinian groups. Their understanding is of interest with respect to the study of more general Kleinian groups. On the other hand, these spaces contain subspaces isomorphic to Teichmüller spaces, and are often useful for the study of properties of Teichmüller space. A recent example of this is the study of the Teichmüller space of the punctured torus by Keen and Series [KS].

We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.