Abstract
The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inversely proportional to the entropy. Using the relation between the two invariants, we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov’s Oka principle in these situations.
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Acknowledgements
The author is grateful to the IHES, the CRM Barcelona, and the Indiana University Bloomington, where a big part of the work on the paper has been done. A talk given at the Algebraic Geometry seminar at Courant Institute and a stimulating discussion with F. Bogomolov and M. Gromov had a special impact. The author is also indebted to many other mathematicians for interesting and helpful discussions and for information concerning references to the literature, among them R. Bryant, D. Calegari, and M. Zaidenberg. The figures were drawn by M. Vergne and F. Dufour.
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To the memory of my colleague Mikael Passare who died tragically in 2011
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Jöricke, B. Gromov’s Oka Principle, Fiber Bundles and the Conformal Module. Acta Math Vietnam 47, 375–440 (2022). https://doi.org/10.1007/s40306-021-00452-z
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DOI: https://doi.org/10.1007/s40306-021-00452-z