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

Abstract: 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 $(\pm\bH)<sup>{r-1}$. We introduce two new ingredients. First, a {multi-dimensional Schwarz-Pick contraction lemma}: the generalized modular embedding $F:\bH\to\bH<sup>{r-1}$, 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 $δ=δ(Γ)&gt;0$ 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 $#(\cL(Γ)\cap[N-1,N])\leq CN<sup>{(r-1)<sup>{3/2}(1-δ)}$ 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} $δ&gt;1-\bigl(\sqrt2\,(r-1)\bigr)<sup>{-1}$, the sharpest threshold our method gives, obtained by a refined geometry-of-numbers argument (Proposition \ref{propsharp}) that improves on the cruder exponent $(r-1)<sup>{3/2}$ obtained directly from the norm form (the two coincide exactly at $r=3$).

• The paper proves exponential growth of mean geodesic-length multiplicities for arbitrary arithmetic dimension when the strong contraction condition δ(Γ) > 1 − 1/[√2(r−1)] holds for r ≥ 3, while recovering unconditional results for r ≤ 2.
• The paper combines multidimensional Schwarz–Pick contraction with geometry-of-numbers norm-form estimates to bound distinct length counts and obtain a growth exponent C ≥ ½[1 − √2(r−1)(1−δ)].
• The results extend to non-cocompact surfaces and identify concrete verification targets, including the arithmetic-dimension-three triangle group Δ(2,3,19), although no example with r ≥ 3 currently satisfies a verified contraction bound.

Context and problem

For a complete finite-area hyperbolic orbifold $S=\Gamma\backslash\mathbb{H}$, the prime geodesic theorem gives $N(\ell)\sim e^\ell/\ell$ for the number of closed geodesics of length at most $\ell$, counted with multiplicity. The mean multiplicity $\langle g(\ell)\rangle$ is defined via a Riemann–Stieltjes relation between $N(\ell)$ and the unweighted counting function $N'(\ell)$, and the surface has exponential growth of mean multiplicities (EGMM) if $\langle g(\ell)\rangle \geq e^{C\ell}$ for some $C>0$. For arithmetic Fuchsian groups, EGMM follows from the bounded clustering property (B-C) of Luo–Sarnak, later shown by Geninska–Leuzinger to characterize arithmeticity in the non-cocompact case. Belolipetsky, Cosac, Dória, and Teixeira Paula extended this to semi-arithmetic groups — those whose invariant trace field $K$ is totally real with algebraic integer traces — admitting modular embeddings, but only for arithmetic dimension $r\leq 2$, explicitly leaving open the case $N(\ell)\sim e^\ell/\ell$0.

The paper under review addresses that question. A semi-arithmetic group of arithmetic dimension $N(\ell)\sim e^\ell/\ell$1 has $N(\ell)\sim e^\ell/\ell$2 non-trivial embeddings $N(\ell)\sim e^\ell/\ell$3 under which the trace set is unbounded, and admits a generalized modular embedding $N(\ell)\sim e^\ell/\ell$4, equivariant under the action of $N(\ell)\sim e^\ell/\ell$5 via these embeddings simultaneously.

Main results

The paper proves two main theorems, in cocompact and non-cocompact form: if $N(\ell)\sim e^\ell/\ell$6 is a semi-arithmetic Fuchsian group of finite covolume, derived from a quaternion algebra, admitting a generalized modular embedding, then $N(\ell)\sim e^\ell/\ell$7 has EGMM whenever either (i) $N(\ell)\sim e^\ell/\ell$8, or (ii) $N(\ell)\sim e^\ell/\ell$9 and $\ell$0 satisfies the strong contraction condition

$\ell$1

where $\ell$2 is the Schwarz–Pick contraction constant of the embedding. The case $\ell$3 recovers the earlier result of BCDT unconditionally; the case $\ell$4 is conditional on an explicit hypothesis that the paper does not verify for any specific group. A unified corollary removes the quaternion-algebra hypothesis at the cost of replacing $\ell$5 (for $\ell$6) and the strong contraction condition (for $\ell$7) by strictly stronger conditions involving $\ell$8; notably, even at $\ell$9 the condition becomes $\langle g(\ell)\rangle$0 when $\langle g(\ell)\rangle$1.

The proof strategy follows the template of the $\langle g(\ell)\rangle$2 case: control the count $\langle g(\ell)\rangle$3 of distinct lengths sub-exponentially, then divide the prime geodesic asymptotic $\langle g(\ell)\rangle$4 by it to obtain $\langle g(\ell)\rangle$5 with $\langle g(\ell)\rangle$6 exactly under the strong contraction condition.

Ingredient I: multi-dimensional Schwarz–Pick contraction

The first new ingredient generalizes the classical Schwarz–Pick argument from one target hyperbolic plane to the product $\langle g(\ell)\rangle$7 with its product Kobayashi metric. Since $\langle g(\ell)\rangle$8 is non-arithmetic, the equivariant map $\langle g(\ell)\rangle$9, $N(\ell)$0, cannot be totally geodesic — otherwise Bergeron–Clozel would force $N(\ell)$1 to be arithmetic. Consequently not every component $N(\ell)$2 is an isometry, and the pointwise equality case of Schwarz–Pick forces the supremum of $N(\ell)$3 over a compact fundamental domain to fall strictly below $N(\ell)$4:

$N(\ell)$5

A key subtlety, made explicit in the paper's remarks, is that for $N(\ell)$6 individual conjugate traces need not contract relative to $N(\ell)$7; only their product does, via Cauchy–Schwarz on the displacement vector. This loss is the source of the exponent gap discussed below.

Ingredient II: geometry-of-numbers norm-form estimate

The second ingredient converts displacement bounds into trace separation. For distinct traces $N(\ell)$8, the norm form satisfies $N(\ell)$9, while each factor is bounded using the contraction estimates: bounded embeddings contribute constants, and the $N'(\ell)$0 unbounded embeddings contribute powers of $N'(\ell)$1. Minkowski-type bookkeeping over the lattice $N'(\ell)$2 yields

$N'(\ell)$3

and hence $N'(\ell)$4. Unlike the case $N'(\ell)$5, this exponent need not be below $N'(\ell)$6 — precisely the obstruction that forced BCDT to stop at $N'(\ell)$7.

A sharpened proposition improves the exponent from $N'(\ell)$8 to $N'(\ell)$9 for $\langle g(\ell)\rangle \geq e^{C\ell}$0 (equal at $\langle g(\ell)\rangle \geq e^{C\ell}$1, strictly better for $\langle g(\ell)\rangle \geq e^{C\ell}$2). The improvement comes from applying Cauchy–Schwarz twice to exploit the joint sum-of-squares constraint on both traces simultaneously rather than bounding each embedding term-by-term. The strong contraction condition is exactly the statement that the sharpened exponent stays below $\langle g(\ell)\rangle \geq e^{C\ell}$3. The paper concedes that neither application of Cauchy–Schwarz is known to be tight for the displacement vectors actually arising from generalized modular embeddings, so further improvement may be possible.

Non-cocompact case

For groups with cusps, the compact-core argument uses the generalized Kirszbraun theorem of Lang–Schroeder, extended here to an equivariant version: a Lipschitz equivariant map defined on the preimage of a compact core extends to all of $\langle g(\ell)\rangle \geq e^{C\ell}$4 with the same Lipschitz constant, constructed orbit-by-orbit via circumcenters in the CAT(0) target $\langle g(\ell)\rangle \geq e^{C\ell}$5. The paper is careful to work on the universal cover rather than the quotient, since local isometry to a Hadamard manifold does not imply global Alexandrov curvature bounds. Combined with a deformation retraction onto the compact core, this yields the same displacement bound with a compact-core constant $\langle g(\ell)\rangle \geq e^{C\ell}$6, and the cocompact proof carries over verbatim; cusps do not affect the prime geodesic leading term.

Examples and the non-vacuousness problem

The paper is candid that for no group of arithmetic dimension $\langle g(\ell)\rangle \geq e^{C\ell}$7 does it verify the strong contraction condition, nor establish that any such group satisfies it. Candidate families are described: totally geodesic curves in Hilbert modular varieties (e.g., over $\langle g(\ell)\rangle \geq e^{C\ell}$8 and $\langle g(\ell)\rangle \geq e^{C\ell}$9), higher-genus Teichmüller curves, and triangle groups. For triangle groups, Cohen–Wolfart guarantees generalized modular embeddings, and Nugent–Voight give finite explicit lists for each $C>0$0 — reducing verification to finitely many computations of $C>0$1 and $C>0$2, none carried out.

The most concrete contribution is the example $C>0$3, the smallest compact triangle group of arithmetic dimension $C>0$4. A direct computation confirms $C>0$5 via Takeuchi's invariant, and identifies the two non-trivial components of the generalized modular embedding as Schwarz triangle maps for angle-triples $C>0$6 and $C>0$7. Verifying the strong contraction condition for this group reduces to bounding derivatives of these hypergeometric maps over a fundamental domain — a well-posed connection problem left unsolved. An attempted numerical estimate produced derivative ratios exceeding the Schwarz–Pick bound, which the authors attribute to a normalization mismatch rather than reporting as an estimate. It therefore remains open whether the class of groups satisfying the strong contraction condition at $C>0$8 is non-empty.

Quantum chaos implications

Via the Bolte–Bogomolny trace-formula heuristic relating length multiplicities to spectral number variance, EGMM is consistent with Poisson-like energy statistics. The exponent satisfies $C>0$9, decreasing in $K$0 for fixed $K$1 and increasing in $K$2; for arithmetic surfaces ($K$3) the maximal value $K$4 is attained. This quantifies the computational observation of Bogomolny–Schmit that multiplicities become harder to detect at larger arithmetic dimension, and explains why the method gives no positive lower bound on $K$5 for groups failing the strong contraction condition. The paper also sketches, without proving, an extension of the framework to the three-dimensional Aurich–Marklof tetrahedral example.

Limitations and open questions

Several limitations are stated plainly. First, the $K$6 results are conditional: the strong contraction condition is sufficient but not claimed necessary, and its non-vacuousness for any specific group is unestablished. Second, the extension beyond quaternion-derived groups requires the stronger condition involving $K$7, and the paper does not know whether this can be weakened. Third, bounding $K$8 requires explicit uniformization data for the relevant Hilbert or Teichmüller curve, which neither this paper nor its predecessors supply. Fourth, the paper notes that generic Fuchsian groups are expected to have bounded mean multiplicity, so EGMM should not be expected for a typical semi-arithmetic group of high dimension. Open questions include: whether the strong contraction condition is automatic for quaternion-derived groups; whether the exponent can be improved to something independent of $K$9 staying below $r\leq 2$0 for all $r\leq 2$1; whether existence of a generalized modular embedding is necessary for EGMM; the sharp value of $r\leq 2$2; and extensions to Kleinian groups and to semi-arithmetic groups without modular embeddings.

Conclusion

This paper extends the theory of exponential growth of mean multiplicities from semi-arithmetic surfaces of arithmetic dimension at most two to arbitrary dimension, at the price of an explicit geometric hypothesis — the strong contraction condition — that holds automatically for $r\leq 2$3 but remains unverified for any group of dimension $r\leq 2$4. Its two technical contributions, the multi-dimensional Schwarz–Pick contraction lemma and the geometry-of-numbers norm-form estimate with its sharpened exponent, reduce the open question of BCDT to concrete, finite, and in principle numerically tractable computations, exemplified by $r\leq 2$5. Whether those computations succeed, or whether counterexamples exist, remains open.