Counting isospectral manifolds

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 xc 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 [2]. Our proof uses Sunada's method, results of [2], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.

Publisher

Elsevier

Publication Date

12-1-2017

Publication Title

Advances in Mathematics

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1016/j.aim.2017.09.040

Keywords

Isospectral manifolds, Counting manifolds, Lattices in semisimple Lie groups

Language

English

Format

text

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