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UBC Theses and Dissertations

Orderable groups and topology Wilmarth, Constance

Abstract

This thesis examines some connections between topology and group theory, in particular the theory of orderable groups. It investigates in close detail some landmark results on this mathematical interface, beginning with Holder's Theorem, and touches upon some recent results in this expanding field of research. Simply stated, Holder's Theorem asserts that Archimedean orderable groups are none other than subgroups of the group of real numbers under addition. Since Holder proved this in 1902, only one significant refinement, due to Paul Conrad, has been made, so these powerful theorems provide the foundation for our understanding of orderable groups. In particular this understanding has served topologists well. This thesis is mostly a distillation of work done in connection with topological applications of the theory, which are surprisingly varied and diverse. Burns and Hale's work on local indicability and right orderability is considered, as well as Bergman's study of the universal covering group of SL(2,R). In addition N. Smythe's extension of a classical result of Alexander's via the left orderability of the fundamental groups of certain surfaces is investigated.

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