Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension
Abstract
For a semi-arithmetic Fuchsian group Γ of arithmetic dimension r with a generalized modular embedding, EGMM holds for r≤2, and for r≥3 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≤2 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.
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BibTeX
@misc{zuevsky2026exponential,
title = {{Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.17604},
eprint = {2608.17604},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17604},
note = {CC0},
}
RIS
TY - UNPB
TI - Exponential Growth of Mean Multiplicities in Length Spectra of Semi-Arithmetic Surfaces of Arbitrary Arithmetic Dimension
AU - A. Zuevsky
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.17604
UR - https://arxiv.org/abs/2608.17604
AB - For a semi-arithmetic Fuchsian group Γ of arithmetic dimension r with a generalized modular embedding, EGMM holds for r≤2, and for r≥3 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≤2 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.
ER -