A Thin Fundamental Set for SL(2, Z)

TL;DR

This paper constructs a thin fundamental set for SL(2, Z) with a controlled K-component and establishes an inequality for the L^2-norm of functions on the quotient space, advancing understanding of the group's geometric and analytical properties.

Contribution

It introduces a new thin fundamental set with a K-component close to the identity and proves an associated L^2-norm inequality for functions on the quotient space.

Findings

Constructed a fundamental set with controlled K-component proximity to identity.

Proved an inequality for the L^2-norm of functions on G/Γ.

Enhanced understanding of geometric structure of SL(2, Z) and its quotient.

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 $G/\Gamma$.