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 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 . Our proof uses Sunada's method, results of , and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.
Belolipetsky, Mikhail, and Benjamin Linowitz. 2017. “Counting isospectral manifolds.” Advances in Mathematics 321: 60-79.
Advances in Mathematics
Isospectral manifolds, Counting manifolds, Lattices in semisimple Lie groups