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 , which are Garside-like half-twists involving strings through , and by counting powered generators as instead of simply . The geometrical complexity is some natural measure of the amount of distortion of the times punctured disk caused by a homeomorphism. Our main...
We investigate the orderability properties of fundamental groups of 3-dimensional
manifolds. Many 3-manifold groups support left-invariant orderings, including all compact
-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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