Kissing Number in Hyperbolic Manifolds

• The paper establishes sharp subquadratic upper bounds for kissing numbers in hyperbolic manifolds by linking systole length with genus or volume.
• It utilizes geometric and combinatorial techniques, such as the collar lemma and trace-length formulas, to rigorously control the clustering of minimal closed geodesics.
• Arithmetic constructions validate these theoretical bounds by producing explicit manifold families with kissing numbers growing as a power of volume under controlled geometric constraints.

The kissing number in hyperbolic manifolds quantifies the number of distinct closed geodesics achieving the minimal possible length (the systole), closely paralleling the classical notion of the kissing number in Euclidean sphere packings. This analogue is essential in the paper of hyperbolic geometry, as it interlinks global topological invariants (such as genus or volume), local geometric features (like systole length), and arithmetic structure. Recent research has established sharp subquadratic upper bounds for hyperbolic surfaces and discovered arithmetic constructions in higher dimensions with kissing numbers exhibiting unexpectedly rapid growth, yet always constrained below certain power laws.

1. Foundational Definitions and Structural Analogies

Let $M$ denote a complete finite-area hyperbolic manifold (including both surfaces and higher-dimensional cases). The systole, $\operatorname{sys}(M)$, is the length of the shortest noncontractible closed geodesic. The kissing number, denoted $\operatorname{Kiss}(M)$, is the number of free homotopy classes of closed geodesics of length exactly $\operatorname{sys}(M)$.

For hyperbolic surfaces of genus $g$, this is formally,

$$\operatorname{Kiss}(S) = \#\{ [\alpha] \neq 0 : \alpha \text{ closed geodesic, } \ell(\alpha) = \operatorname{sys}(S)\}.$$

This definition abstracts the classical sphere kissing arrangement to the setting where "spheres" are replaced by systolic geodesics. In higher dimensions, the same philosophy holds: shortest closed geodesics form the "touching" configuration, constrained by the geometry and topology of $M$.

2. Upper and Lower Bounds: Genus and Volume Growth

Hyperbolic Surfaces ($n=2$)

Schmutz Schaller constructed families of arithmetic hyperbolic surfaces exhibiting large kissing numbers. Specifically, the maximal kissing number for a surface of genus $g$ satisfies

$\operatorname{Kiss}(S) \geq g^{4/3-\epsilon}$

for any $\epsilon>0$ (Parlier, 2011, Murillo, 31 Aug 2025). He further conjectured the matching upper bound $\operatorname{Kiss}(S) \leq B g^{4/3}$ for some universal $B$.

Explicit subquadratic upper bounds have been established:

• For a surface $S$ of genus $g$ and systole length $\ell$,

$$\operatorname{Kiss}(S) \leq U \cdot \frac{e^{\ell/2}}{\ell} \cdot g$$

where $U$ is an absolute constant (explicitly, $U=100$) (Parlier, 2011).

• Since $\ell \lesssim 2 \log g$, this yields

$\operatorname{Kiss}(S) \leq C \frac{g^2}{\log g}$

with $C$ explicit and universal (Fanoni et al., 2014).

• For finite-area surfaces of signature $(g,n)$,

$$\operatorname{Kiss}(S) \leq C (g + n) \frac{g}{\log(g+1)}$$

(with $C = 2 \times 10^4$) (Fanoni et al., 2014), applicable when $n$ (cusps) is fixed or grows linearly with $g$.

These bounds are sharp up to (poly)logarithmic factors, showing that the systolic constraint and genus lead to subquadratic growth in contrast with naive combinatorial estimates.

Higher-Dimensional Hyperbolic Manifolds ($n \geq 3$)

For closed hyperbolic $n$-manifolds, the kissing number, i.e., the number of systolic geodesics, is bounded by

$$\operatorname{Kiss}(M) \leq A_n \cdot \operatorname{vol}(M) \cdot \frac{e^{(n-1)\operatorname{sys}(M)/2}}{\operatorname{sys}(M)}$$

for a constant $A_n$ depending only on the dimension (Bourque et al., 2019). Using a fundamental volume-systole relation,

$$\operatorname{sys}(M) \leq \frac{2}{n-1} \log(\operatorname{vol}(M)) + c$$

this leads to the global subquadratic estimate

$$\operatorname{Kiss}(M) \leq C \frac{\operatorname{vol}(M)^2}{\log(1+\operatorname{vol}(M))}$$

with $C$ universal and explicit (Bourque et al., 2019, Murillo, 31 Aug 2025).

Arithmetic constructions give examples with

$$\operatorname{Kiss}(M_i) \geq \operatorname{vol}(M_i)^{\frac{31}{27}-\epsilon}, \quad \forall \epsilon>0$$

as shown in explicit sequences of noncompact finite volume 3-manifolds built using congruence subgroups of Bianchi groups (Dória et al., 2020). In dimension 2, similar phenomena yield

$$\operatorname{Kiss}(S_i) \geq (\operatorname{area}(S_i))^{4/3-\epsilon}$$

and

$$\operatorname{sys}(S_i) \geq \frac{4}{3}\log(\operatorname{area}(S_i)) - c$$

(Murillo, 31 Aug 2025).

3. Geometric and Combinatorial Techniques Underlying the Bounds

Systolic and Angular Separation

Bounding the kissing number requires strong geometric control over how many systolic geodesics can “cluster” in a small area.

• Collar Lemma: The neighborhood of a systole is a collar of explicit width, and disjoint systoles are separated by a uniform distance (function of $\ell$).
• For two systoles intersecting a disk of radius $r(\ell) \simeq 1 / (2 \sinh(\ell/4))$, their intersection angle $\theta$ satisfies

$\sin \theta \geq \frac{1}{2 \cosh(\ell/4)}$

limiting how many systoles can cross the same region (Parlier, 2011).

• Covering the surface with such disks, counting the number of systoles passing through each, and using angular separation arguments, yields the main geometric ingredient for the universal upper bounds.

Intersection Patterns

In surfaces with cusps, systoles can intersect up to twice (if one curve bounds two cusps). Systoles are divided into classes based on intersection properties (classes $A(S)$, $B(S)$, $C(S)$ in (Fanoni et al., 2014)), culminating in a fine-grained combinatorial analysis that sharpens the overall counting argument.

Trace-Length Formulas and Arithmetic Subgroup Constructions

For arithmetic manifolds, explicit relationships between the traces of matrix representatives in $\mathrm{SL}_2$ over rings of integers and the lengths of geodesics allow one to both produce many conjugacy classes (primitive closed geodesics of the same length) and control their length:

$$\cosh(\ell_\gamma/2) = \frac{|\operatorname{tr}(\gamma)|}{2}$$

(Murillo, 31 Aug 2025). Higher-dimensional analogues exist using loxodromic elements of the corresponding arithmetic groups.

4. Sphere Packing Analogues and the Role of Curvature

The analogy with classical kissings numbers for sphere packings and spherical codes is deep but nontrivial in constant negative curvature.

• In hyperbolic $n$-space, the “kissing function” $\kappa(n, r)$ counts maximal arrangements of mutually tangent radius-$r$ balls around a central ball. Negative curvature causes exponential versus polynomial volume growth; as a result,

$\kappa(n, r) \sim c_n \, e^{(n-1) r}$

as $r \to \infty$, where $c_n$ involves sphere packing density in $\mathbb{R}^{n-1}$ and the Beta function (Dostert et al., 2020, Dostert et al., 2019).

The adaptation of linear programming and recent semidefinite programming (SDP) bounds for spherical codes to hyperbolic space is nontrivial due to the difference in harmonic analysis; the positive-definite kernel constructions and SDP relaxations must be tailored to reflect angular or distance constraints in hyperbolic metric (Boyvalenkov et al., 2015, Dostert et al., 2019, Dostert et al., 2020). Numeric SDP optimization improves upper bounds and in some cases approaches the known lower bounds from explicit constructions.

5. Arithmetic Manifolds with Extremal Kissing Number

Arithmetic constructions provide explicit manifolds with large kissing numbers and controlled systoles. The methodology is as follows:

• Start with a number field and its ring of integers (e.g., $\mathcal{O}_k$ for $k=\mathbb{Q}(\sqrt{-d})$, $d>0$ squarefree with class number 1).
• Form congruence subgroups of $\mathrm{SL}_2(\mathcal{O}_k)$ or its higher-dimensional analogs.
• Use binary quadratic forms and Pell-type equations to control the lengths of geodesics and their count.
• Count equivalence classes to tally the number of systolic geodesics. For suitable principal congruence coverings, the systole grows logarithmically with the volume, and the number of systolic geodesics grows as a power of the volume (e.g., exponent $31/27$ in 3D (Dória et al., 2020)), analogous to $4/3$ in 2D (Murillo, 31 Aug 2025).

This control relies critically on trace relations and class number estimates from analytic number theory.

6. Open Problems and Research Directions

Despite strong progress, several questions remain open:

• The sharpness of exponents: Is the $4/3$ exponent for the genus in surfaces and $31/27$ for the volume in 3-manifolds optimal for all hyperbolic manifolds, or only for arithmetic constructions (Murillo, 31 Aug 2025)?
• Non-arithmetic examples: Can non-arithmetic or random constructions achieve similar kissing numbers, or are such extremal behaviors unique to arithmetic settings?
• Extension beyond hyperbolic space: Can one obtain analogous kissing number estimates for other locally symmetric spaces, not of constant negative curvature?
• Precise constants: Optimizing the constants in inequalities relating $\operatorname{Kiss}(M)$, $\operatorname{sys}(M)$, and $\operatorname{vol}(M)$.

A plausible implication is that the rigidity inherent in hyperbolic geometry (versus purely topological or general Riemannian settings) is both a constraint and a source of structure, leading to subquadratic (or sub-power law) upper bounds on the kissing number. The interplay between arithmetic, geometry, and spectral invariants is central to further advances.

7. Summary Table: Key Results on Kissing Number Growth

Setting	Lower Bound (Known Examples)	Universal Upper Bound	Asymptotic/Exponent
Hyperbolic surfaces ($g$)	$g^{4/3-\epsilon}$	$C \frac{g^2}{\log g}$	Subquadratic in $g$
Hyperbolic 3-manifolds	$\operatorname{vol}^{31/27-\epsilon}$	$$C \frac{\operatorname{vol}^2}{\log \operatorname{vol}}$$	Subquadratic in $\operatorname{vol}$
Hyperbolic $\mathbb{H}^n$, balls ($r$)	$c e^{(n-1)r}$	$C e^{(n-1)r}$	Exponential in $r$

This table encapsulates the core contrast: specific (often arithmetic) constructions attain the best known lower bounds, while universal geometric or analytic arguments yield matching subquadratic or exponential upper bounds up to logarithmic or constant factors.

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The paper of kissing number phenomena in hyperbolic manifolds thus bridges low-dimensional topology, differential geometry, spectral theory, and arithmetic group theory, yielding precise estimates that reveal both the power and constraints of negative curvature and arithmetic structure (Parlier, 2011, Fanoni et al., 2014, Dória et al., 2020, Bourque et al., 2019, Dostert et al., 2020, Murillo, 31 Aug 2025).