Albert Marden

The monodromy groups of Schwarzian equations on closed Riemann surfaces

Let θ : π1(R) → PSL(2, ℂ) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, ℂ).

Schottky Groups and Circles

This chapter discusses schotty groups and circles. Let α 1 , β 1 , … , α g , β g be a set of 2g, g ≥ 2, mutually disjoint Jordan curves on the Riemann sphere S 2 , which comprise the boundary of a 2 g -connected region Ω. If there are g Mobius transformations A 1 , … , A g , which have the property that A i maps α i into β i and A i (Ω) ∩Ω = o, 1 ≤ i ≤ g. Then, the g necessarily loxodromic (this term includes hyperbolic) transformations A i generate what is called a Schottky group of genus g with Ω as a fundamental region. The chapter presents a proof of the theorem ρ0−∩ ρ ≠ ρ, where ρ0-denotes the closure of ρ 0 in V .

Complex projective structures on Kleinian groups

Let M be a compact, oriented, irreducible, and boundary incompressible 3–manifold. Assume that its fundamental group is without rank two abelian subgroups and ∂M 6= ∅. We will show that every homomorphism θ: π1(M )→ PSL(2,C) which is not “boundary elementary” is induced by a possibly branched complex projective structure on the boundary of a hyperbolic manifold homeomorphic to M . AMS Classification 30F50; 30F45, 30F60, 30F99, 30C99