Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension
Year: 2026
Type: preprint
Abstract: For a semi-arithmetic Fuchsian group $Γ$ 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 $δ$ for the latter via its Lyapunov exponents ($δ=2/3$ on ergodic average, by Bainbridge exact computation), explaining why an average bound does not yet certify the pointwise one \eqref{eqstrong-contraction} needs.
Source: arXiv (Cornell University)
Author: Zuevsky, A.
Institution: Czech Academy of Sciences, Institute of Mathematics
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Akademie Věd České Republiky
Awards: G1741043573, G3967893287
OpenAlex ID: W7204216748