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On the complexity of braids

Ivan Dynnikov;  Bert Wiest — 2007

Journal of the European Mathematical Society

We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators i j , which are Garside-like half-twists involving strings i through j , and by counting powered generators Δ i j k as log ( | k | + 1 ) instead of simply | k | . The geometrical complexity is some natural measure of the amount of distortion of the n times punctured disk caused by a homeomorphism. Our main...

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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