Free Group

Uniformization 


A Schottky group is a group \( \Gamma \) with generators \( \gamma_{1}, \ldots, \gamma_{p}, p \geq 1, \) such that there exist \( 2 p \) disjoint Jordan curves \( l_{1}, l_{1}^{\prime}, \ldots, l_{p}, l_{p}^{\prime} \) bounding a \( 2 p- \) connected region \( D \) for which \( \gamma_{j}(D) \cap D=\varnothing \) and \( \gamma_{j}\left(l_{j}\right)=l_{j}^{\prime}, j=1, \ldots, p . \) The group \( \Gamma \) is necessarily free and purely loxodromic, i.e., all its elements \( \gamma \in \Gamma \backslash\{I\} \) are loxodromic or hyperbolic. The factor \( \Omega(\Gamma) / \Gamma \) is a closed surface of genus \( p . \)

The geometric approach to Schottky groups is based on the concept of a fundamental region. A fundamental region of a discontinuous group \( \Gamma \) is defined to be a set \( F \subset \Omega(\Gamma) \) containing one point from each orbit \( \Gamma z_{0}, z_{0} \in \Omega(\Gamma), \) and such that each nonempty component \( F \cap \Omega \) of it is connected.

The Koebe uniformization theorem asserts that all closed Riemann surfaces can be uniformized by the  Schottky manifestation of free groups; for an appropriate choice of a canonical system of generators \( a_{1}, b_{1}, \ldots, a_{p}, b_{p} \) of the fundamental group \( \pi_{1}(\Omega(\Gamma) / \Gamma), \) the covering \( \pi \) here corresponds to the smallest normal subgroup containing \( a_{1}, \ldots, a_{p} \). Conclusively, all closed Riemann surfaces are uniformizable by the concomitant action of a free and loxodromic group. 

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