论文标题
$ 3 $编织和计数功能的保形不变式
Conformal invariants of $3$-Braids and Counting Functions
论文作者
论文摘要
我们考虑了辫子的形状不变的，这是具有完全真实的水平边界值$λ_{tr} $的极端长度。不变的降低到$ \ Mathcal {b} _n \ diagup \ diagup \ Mathcal {z} _n $的元素的不变性，编织组模拟其中心。我们证明$ \ Mathcal {B} _3 \ diagup \ Mathcal {z} _3 $的元素数量正$λ_{tr} $成倍增长。该估计值适用于以几何shafarevich的精神获得有效的有限定理，而在第二类的黎曼表面上。作为推论，我们获得了$ \ Mathcal {b} _3 \ diagup \ Mathcal {z} _3 $的另一个结合类数量的指数增长的证明，其正熵不超过$ y $。
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 $\mathcal{B}_n\diagup\mathcal{Z}_n$, the braid group modulo its center. We prove that the number of elements of $\mathcal{B}_3\diagup\mathcal{Z}_3$ 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 $\mathcal{B}_3\diagup\mathcal{Z}_3$ with positive entropy not exceeding $Y$.
