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The topology on the space of left orderings of a group

Peter A. Linnell
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Abstract

Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to answer three conjectures. In particular we shall show that O_G can never be countably infinite. Furthermore in the case G is a countable nonabelian free group, we shall show that O_G is homeomorphic to the Cantor set and that the positive cone of a left order on G is not finitely generated. Generalizations to locally indicable groups will also be considered.

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Peter A. Linnell. 2006-07-24. The topology on the space of left orderings of a group. https://arxiv.org/abs/math/0607470

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