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A topological view of ordered groups

Dale Rolfsen — 2014

Banach Center Publications

In this expository article we use topological ideas, notably compactness, to establish certain basic properties of orderable groups. Many of the properties we shall discuss are well-known, but I believe some of the proofs are new. These will be used, in turn, to prove some orderability results, including the left-orderability of the group of PL homeomorphisms of a surface with boundary, which are fixed on at least one boundary component.

Geometric subgroups of surface braid groups

Luis Paris;  Dale Rolfsen — 1999

Annales de l'institut Fourier

Let M be a surface, let N be a subsurface, and let n ≤ m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group B m M with m strings on M . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of π 1 N in π 1 M . Then we calculate the commensurator, the normalizer and the centralizer of B n N in B m M for large surface braid...

Orderable 3-manifold groups

Steven Boyer;  Dale Rolfsen;  Bert Wiest — 2005

Annales de l’institut Fourier

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P 2 -irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize which manifolds’ groups support a left-invariant or bi-invariant ordering. We also show that manifolds modelled on these geometries have virtually bi-orderable groups. The question of virtual orderability...

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