Counting isospectral manifolds

TL;DR

This paper establishes a lower bound on the number of isospectral non-isometric locally symmetric spaces associated with certain Lie groups, revealing rapid growth rates and connecting to deep number theory results.

Contribution

It provides the first significant lower bound on the count of isospectral manifolds for higher rank Lie groups, approaching known total estimates and utilizing advanced number theory techniques.

Findings

Lower bound grows as x^{c log x / (log log x)^2}

Bound approaches total number of such spaces in higher rank cases

Uses Sunada's method and deep number theory results

Abstract

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [BL]. Our proof uses Sunada's method, results of [BL], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.