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Complex projective structures on Kleinian groups

### Albert Marden

Geometry & Topology Monographs 1 (1998) 335–340
DOI: 10.2140/gtm.1998.1.335
 arXiv: math.GT/9810196
##### Abstract
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Let ${M}^{3}$ be a compact, oriented, irreducible, and boundary incompressible $3$–manifold. Assume that its fundamental group is without rank two abelian subgroups and $\partial {M}^{3}\ne \varnothing$. We will show that every homomorphism $\theta :\phantom{\rule{0.3em}{0ex}}{\pi }_{1}\left({M}^{3}\right)\to PSL\left(2,C\right)$ which is not “boundary elementary” is induced by a possibly branched complex projective structure on the boundary of a hyperbolic manifold homeomorphic to ${M}^{3}$.

##### Keywords
projective structures on Riemann surfaces, hyperbolic 3–manifolds
##### Mathematical Subject Classification
Primary: 30F50
Secondary: 30F45, 30F60, 30F99, 30C99