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Fundamental groups, 3-braids, and effective estimates of invariants

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Jöricke,  Burglind
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Jöricke, B. (2020). Fundamental groups, 3-braids, and effective estimates of invariants. Mathematische Zeitschrift, 294(3-4), 1553-1609. doi:10.1007/s00209-019-02317-6.


Cite as: https://hdl.handle.net/21.11116/0000-0006-0C44-1
Abstract
We define invariants of braids rather than invariants of conjugacy classes of braids. For any pure three-braid we give effective upper and lower bounds for these invariants. This is done in terms of a natural syllable decomposition of the word representing the image of the braid in the braid group modulo its center. The bounds differ by a multiplicative constant not depending on the word. Respective bounds are given for all three-braids. We also obtain effective upper and lower bounds for the entropy of pure three-braids in these terms. The proof leads to the study of the extremal length of classes of curves
representing elements of the fundamental group of the twice punctured complex plane.