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On the Cantor-Bendixson rank of metabelian groups

Yves Cornulier (2011)

Annales de l’institut Fourier

We study the Cantor-Bendixson rank of metabelian and virtually metabelian groups in the space of marked groups, and in particular, we exhibit a sequence ( G n ) of 2-generated, finitely presented, virtually metabelian groups of Cantor-Bendixson rank  ω n .

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...

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