Counting isospectral manifolds

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(clogx/(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. (c) 2017 Elsevier Inc. All rights remserved.