An $\Omega$-Result for the Counting of Geodesic Segments in the Hyperbolic Plane

TL;DR

This paper establishes a lower bound on the error term in counting geodesic segments near a fixed geodesic in the hyperbolic plane, revealing the limitations of existing asymptotic estimates.

Contribution

It provides the first $ ext{Omega}$-result for the error term in the asymptotic counting of geodesic segments in hyperbolic geometry.

Findings

Error term has a lower bound of order $X^{1/2}( ext{log log } X)^{1/4- ext{delta}}$

Demonstrates limitations of asymptotic counting methods in hyperbolic geometry

Advances understanding of geodesic distribution in hyperbolic surfaces

Abstract

Let $\Gamma$ be a cocompact Fuchsian group, and $l$ a fixed closed geodesic. We study the counting of those images of $l$ that have a distance from $l$ less than or equal to $R$. We prove an $\Omega$-result for the error term in the asymptotic expansion of the counting function. More specifically, we prove that the error term is equal to $\Omega_{\delta}\left(X^{1/2}\left(\log{\log{X}}\right)^{1/4-\delta} \right)$, where $X=\cosh{R}$.