Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension

Abstract

For a semi-arithmetic Fuchsian group $\Gamma$ of arithmetic dimension $r$ with a generalized modular embedding, EGMM holds for $r\leq2$, and for $r\geq3$ under an explicit \emph{strong contraction condition}, via a multi-dimensional Schwarz-Pick bound and a geometry-of-numbers estimate. A covering construction turns every known example into an infinite explicit family with the same exponent via principal congruence towers. We add new $r\leq2$ examples (genus-$3$/$4$ Prym eigenform loci, a different commensurability class from the known triangle groups and McMullen genus-two family) and attempt a direct flat-geometric bound on the contraction constant $\delta$ for the latter via its Lyapunov exponents ($\delta=2/3$ on ergodic average, by Bainbridge exact computation), explaining why an average bound does not yet certify the pointwise one \eqref{eqstrong-contraction} needs.