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The University of Southampton
Mathematical Sciences

Research project: Schottky groups and uniformizations of closed surfaces

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Schottky groups are a class of Kleinian groups which admit a particularly simple construction; namely, remove 2g (pairwise disjoint) open discs bounded by Jordan curves from the Riemann sphere and pair the boundary curves of the resulting region by Moebius transformations.

Such a Schottky group is a free group of rank g, uniformizing a closed Riemann surface of genus g. Chuckrow (1968) showed, among other results, that this construction applies to every free basis for a Schottky group G. Marden (1974) demonstrated that there are two distinct types of Schottky groups, namely those for which the curves for some free basis can be taken to be circles (these are the classical Schottky groups) and those for which no set of such curves for any free basis can be taken to be circles (these are the non-classical Schottky groups).

The goal of this project is to consider the question arising straightforwardly from Marden's proof of the existence of non-classical Schottky groups, and which remains unanswered in full generality: Does every closed Riemann surface admit a uniformization by a classical Schottky group?

Papers: Subgroups of classical and non-classical Schottky groups (in preparation)

Related research groups

Pure Mathematics
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