Braids, Conformal Module and Entropy

TL;DR

This paper explores the relationship between the conformal module of braid conjugacy classes and the entropy invariant, revealing an inverse proportionality and applying these concepts to algebraic geometry.

Contribution

It establishes a connection between conformal module and entropy for braids and demonstrates their application in algebraic geometry.

Findings

Conformal module is inversely proportional to braid entropy.

The relationship links geometric and dynamical braid invariants.

Application to algebraic geometry illustrates broader implications.

Abstract

The conformal module of conjugacy classes of braids implicitly appeared in a paper of Lin and Gorin in connection with their interest in the 13. Hilbert Problem. This invariant is the supremum of conformal modules (in the sense of Ahlfors) of certain annuli related to the conjugacy class. This note states that the conformal module is inverse proportional to a popular dynamical braid invariant, the entropy. The entropy appeared in connection with Thurston's theory of surface homeomorphisms. An application of the concept of conformal module to algebraic geometry is given.