The Hausdorff Dimension of Limit Sets of Well-distributed Schottky Groups
By: William Huanshan Chuang
Department: Mathematics
Faculty Advisors: Dr. Chun-Kit Lai, Dr. Emily Clader, and Dr. Dustin Ross
Let a finitely generated Schottky group $G$ be given, and $\mathbb{B}^2$ be a Poincar\'{e} disk model of two-dimensional hyperbolic space. An unsolved problem is: How to find an exact value of Hausdorff dimension of the limit set $L(G)$ when $L(G)\neq \partial \mathbb{B}^2$?
To solve one specific case of this problem, a well-distributed Schottky group $\Gamma=\left\langle T_1,T_2,...,T_m\right\rangle$ is defined. Our main theorem gives sharp bounds on the critical exponent of Poincar\'{e} series, and the theorem was proved based on: properties of Poincar\'{e} series, isometric circles, and some nice properties come with the definition of well-distributed Schottky group, especially that we found a way to reconstruct the orbit $\Gamma(0)$ by using only two operators $T$ and $R$ for any $m\in\mathbb{Z}^+$. Following the proof of the main theorem, for the first time, an exact form of Hausdorff dimension for all possible well-distributed Schottky groups of rank two was conjectured and used to generate results against the best approximation derived using McMullen's algorithm.