Correlation between the algebraic length of words in a Fuchsian fundamental group and the geometric length of their corresponding closed geodesics
2018
Let $S=\Gamma\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $\Gamma \le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary. We prove that there exists a generating set $\mathfrak S$ of $\Gamma$ such that when sampling length-$n$ words built from the elements of $\mathfrak S$ as $n\to \infty$, the subset of this sampled set comprised of words that are hyperbolic in $\pi_1(S)\cong \Gamma$ approaches full measure, and the distribution of geometric lengths of the closed geodesics corresponding to words in this subset converges (when normalized) to a Gaussian. In addition to this Central Limit Theorem, we also show a Law of Large Numbers, Law of the Iterated Logarithm, Large Deviations Principle, and Local Limit Theorem for this distribution.
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