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

We study the exponential growth of mean multiplicities (EGMM) in the geodesic length spectrum of a semi-arithmetic Fuchsian group $Γ$ of finite covolume and arithmetic dimension $r\geq 1$ admitting a generalized modular embedding into $(\pm\bH)^{r-1}$. We introduce two new ingredients. First, a {multi-dimensional Schwarz-Pick contraction lemma}: the generalized modular embedding $F:\bH\to\bH^{r-1}$, being holomorphic and strictly contracting with respect to the product Kobayashi metric, satisfies $\norm{DF_z}_{\mathrm{op}}\leq\sqrt{r-1}\,(1-δ)$ for a uniform $δ=δ(Γ)>0$ and all $z\in\bH$. Second, a geometry-of-numbers norm-form estimate: Minkowski's theorem applied to the lattice of algebraic integers in the invariant trace field $K$ gives $\#(\cL(Γ)\cap[N-1,N])\leq CN^{(r-1)^{3/2}(1-δ)}$ for all $N$; unlike the case $r\leq 2$, this exponent depends on $r$. Combining the two ingredients shows that $Γ$ has EGMM whenever $r\leq 2$, or $r\geq 3$ and $δ$ satisfies the {strong contraction condition} $δ>1-\bigl(\sqrt2\,(r-1)\bigr)^{-1}$, the sharpest threshold our method gives, obtained by a refined geometry-of-numbers argument (Proposition \ref{propsharp}) that improves on the cruder exponent $(r-1)^{3/2}$ obtained directly from the norm form (the two coincide exactly at $r=3$).