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no. 5
Conformal Invariants of 3-Braids and Counting Functions
Burglind Jöricke1
1 IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1323-1341.

We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values Λ tr . The invariant descends to an invariant of elements of ℬ n ∕𝒵 n , the braid group modulo its center. We prove that the number of elements of ℬ 3 ∕𝒵 3 of positive Λ tr grows exponentially. The estimate applies to obtain effective finiteness theorems in the spirit of the geometric Shafarevich conjecture over Riemann surfaces of second kind. As a corollary we obtain another proof of the exponential growth of the number of conjugacy classes of ℬ 3 ∕𝒵 3 with positive entropy not exceeding Y.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1721
Classification: 30F60, 30CXX, 37B40, 57MXX, 20F36
Keywords: Braids, extremal length, mapping classes, entropy, counting function

Burglind Jöricke 1

1 IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Burglind Jöricke. Conformal Invariants of 3-Braids and Counting Functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 31 (2022) no. 5, pp. 1323-1341. doi : 10.5802/afst.1721. https://afst.centre-mersenne.org/articles/10.5802/afst.1721/

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