The super-polynomial lower-bound conjecture for type A1 semisimple Lie groups

Let $H$ be a semisimple Lie group of type $\mathrm{A}_1$ with more than one simple factor, and let $\mathrm{IL}_{H}(x)$ denote the maximal size of a family of pairwise isospectral, non-isometric manifolds associated with $H$ as defined in the paper. Theorem~ gives a lower bound of the form $\mathrm{IL}_{H}(x) \ge x^{c\log x/ (\log\log x)^2}$ for the semisimple groups covered by its hypotheses. Type $\mathrm{A}_1$ lower-bound conjecture. The lower bound of Theorem~ holds for semisimple Lie groups of type $\mathrm{A}_1$ which have more than one simple factor. The paper's main theorem proves the analogous bound for almost all non-compact semisimple Lie groups, but explicitly leaves these higher-rank exceptions unresolved; this claim is presented as a desired extension rather than an established result.