A Thin Fundamental Set for SL(2, Z)

Abstract

Let $\Gamma =SL(2,\mathbb{Z})$ and $G=SL(2,\mathbb{R})$. Let $g=kan$ be the Iwasawa decomposition. Let $\epsilon$ be a small positive number. In this paper, we construct a fundamental set ${\mathcal{F}}_{\epsilon }$ such that the $k$-component of $g\in {\mathcal{F}}_{\epsilon }$ is within the $\epsilon$-distance from the identity. We further prove an inequality for the $L^2$-norm of functions on $G/\Gamma$.

11 pages