Conformal invariants of $3$-Braids and Counting Functions

Abstract

We consider a conformal invariant of braids, the extremal length with totally real horizontal boundary values ${\lambda }_{tr}$. The invariant descends to an invariant of elements of ${\mathcal{B}}_n╱{\mathcal{Z}}_n$, the braid group modulo its center. We prove that the number of elements of ${\mathcal{B}}_3╱{\mathcal{Z}}_3$ of positive ${\lambda }_{tr}$ 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 ${\mathcal{B}}_3╱{\mathcal{Z}}_3$ with positive entropy not exceeding $Y$.

arXiv commentpaper revised, a mistake corrected. arXiv admin note: substantial text overlap with arXiv:1412.7000