On Lifting Kleinian Groups to SL (2, ℂ)

Irwin Kra

A chapter in Differential Geometry and Complex Analysis, 1985, pp 181-193 from Springer

Abstract: Abstract The exact sequence of groups and group homomorphisms (0.1) $$1 \to \{ \pm I\} \to SL(2,\mathbb{C})\xrightarrow{\mathcal{P}}PSL(2,\mathbb{C}) \to 1$$ does not split. If in this sequence we identify PSL(2, ℂ) with the Möbius group, then for $$A = \left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right) \in SL(2,\mathbb{C}),\mathcal{P}(A)$$ is the Möbius transformation $$z \mapsto \frac{{az + b}}{{cz + d}}$$ . A lift of an element α ∈ PSL (2, ℂ) is an element A ∈ SL (2, ℂ) with P (A) = α, while a lift of a subgroup Γ of PSL(2, ℂ) is an isomorphism i: Γ → SL(2, ℂ) such that P ° i is the identity.

Keywords: Riemann Surface; Parabolic Subgroup; Automorphic Form; Kleinian Group; Fuchsian Group (search for similar items in EconPapers)
Date: 1985
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DOI: 10.1007/978-3-642-69828-6_13

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