Temperedness of locally symmetric spaces: The product case

TL;DR

This paper investigates the spectral properties of quotients of product symmetric spaces, showing that under certain conditions, the associated $L^2$ spaces are tempered, linking spectral theory with geometric growth in the product setting.

Contribution

It establishes a connection between the spectrum of product symmetric spaces and the asymptotic growth of discrete subgroups, demonstrating temperedness for a broad class of groups.

Findings

Spectrum of $ ext{Gamma} ackslash X$ relates to subgroup growth in each factor.

$L^2( ext{Gamma} ackslash G)$ is tempered for many $ ext{Gamma}$.

Provides new insights into the harmonic analysis on product symmetric spaces.

Abstract

Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $\Gamma$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $\Gamma \backslash X$ is related to the asymptotic growth of $\Gamma$ in the two direction defined by the two factors. We obtain that $L^2(\Gamma \backslash G)$ is tempered for large class of $\Gamma$.