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

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 $X=\cosh R$.

14 pages. Comments welcome!