Arithmetic Manifolds in Geometry & Number Theory

• Arithmetic manifolds are locally symmetric spaces constructed as quotients of symmetric spaces by arithmetic groups, linking geometric, arithmetic, and spectral theories.
• They are built from discrete group actions via structures like quaternion algebras and orthogonal groups, yielding examples such as hyperbolic manifolds with finite volume.
• Research on arithmetic manifolds uncovers deep connections among systolic geometry, spectral invariants, analytic torsion, and topological features like cusp structures and cohomology torsion.

An arithmetic manifold is a locally symmetric space, typically of non-compact type (most commonly hyperbolic, but also more generally associated to semisimple Lie groups), that is constructed as a quotient of the associated symmetric space by an arithmetic group—in particular, a discrete subgroup defined via the integral (or otherwise arithmetically meaningful) points of an algebraic group over a number field. The ubiquity and importance of arithmetic manifolds in modern geometry, topology, and number theory stem from their rich structure, which encodes interactions between geometry, arithmetic, automorphic forms, and spectral theory. The canonical examples are arithmetic hyperbolic manifolds in dimensions 2 and higher, constructed via quaternion algebras or orthogonal groups over number fields.

1. Algebraic Construction and Definitions

Arithmetic manifolds arise from lattice quotients of symmetric spaces G/K by arithmetic subgroups Γ ≤ G(ℝ), where G is a semisimple algebraic group defined over ℚ or a number field F and K is a maximal compact subgroup. The prototypical constructions are as follows:

• Orthogonal group case: Fix a number field F (often totally real for hyperbolic manifolds), let q be a quadratic form over F of suitable signature (typically (n,1) under a chosen real embedding and positive-definite on the others). The associated real Lie group is Isom⁺(Hⁿ) ≅ SO⁺(n,1). Taking Γ = SO⁺(q;𝒪_F), the group of integral points with respect to some structure (possibly maximal order), the quotient Hⁿ/Γ is a finite volume hyperbolic manifold or orbifold.
• Quaternion algebra case: For hyperbolic 2- and 3-manifolds, arithmetic subgroups of PSL₂(ℝ) and PSL₂(ℂ) can be constructed using norm-one elements (modulo center) in maximal orders of quaternion algebras defined over totally real or imaginary quadratic fields (see (Murillo, 31 Aug 2025, Bergeron et al., 2014, Heck et al., 2022)).
• Unitary and higher-rank cases: More general arithmetic manifolds—for example, quotients of SL(n,ℝ)/SO(n)—are obtained by lattices defined by unitary groups SU(J,𝒪_F,τ) where J is a τ-Hermitian form over a number field F with Galois automorphism τ, or by restricting integral/𝒪_F–points of G embedded by an admissible algebraic representation (Hillen, 21 Mar 2024).
• Definitions: A locally symmetric manifold M is called arithmetic if it is commensurable with a quotient X/Γ, where X is the symmetric space associated to G and Γ is an arithmetic lattice as above (see (Murillo, 31 Aug 2025, Kolpakov et al., 2017)).

The commensurability class (equivalence up to passing to finite-index subgroups) of an arithmetic manifold is governed by the isomorphism class (or more subtly, the projective equivalence) of the underlying algebraic structure ("defining quadratic form," quaternion algebra, etc.) and, in many applications, by number theoretic invariants such as discriminant, trace field, or ramification data (McCoy et al., 14 Oct 2024, Linowitz, 2017).

2. Geometric and Spectral Invariants: Systole, Volume, Length Spectrum

Arithmetic manifolds are central objects in the paper of relationships between geometric and topological invariants, such as volume, systole (shortest closed geodesic), length/kissing spectrum, injectivity radius, and the growth of torsion in (co)homology and regulator invariants.

• Systole–Volume Relations: There are logarithmic lower bounds on the systole of congruence arithmetic manifolds in both low and higher dimensions. For hyperbolic surfaces,

$$\mathrm{sys}(S_N) \gtrsim \frac{4}{3}\log(\mathrm{area}(S_N)) - c$$

for congruence covers $S_N$ of the modular surface, and similar results hold in higher dimensions with constants depending on $n$ (Murillo, 31 Aug 2025, Heck et al., 2022).

• Kissing Number: The number of closed geodesics of minimal length grows polynomially with volume. Explicit estimates in dimension 2 yield, for instance,

$$\mathrm{Kiss}(S_N) \gtrsim (\mathrm{area}(S_N))^{4/3 - \varepsilon}$$

for congruence coverings (Murillo, 31 Aug 2025).

• Length Spectrum Arithmetic Progressions: For non-compact locally symmetric arithmetic manifolds, the length spectrum contains arbitrarily long arithmetic progressions, a phenomenon that is highly non-generic for negatively curved metrics. The presence of such progressions, and even the patterning of all primitive geodesic lengths into progressions, is a spectral signature of arithmeticity (Lafont et al., 2014).
• Volume–Systole Construction: There exist infinite families of non-commensurable arithmetic manifolds with the same systole, constructed by altering ramification sets in quaternion algebras while preserving the embedding of a fixed quadratic extension (Heck et al., 2022).
• Systole in Higher Rank: In certain higher-rank arithmetic locally symmetric spaces, it is possible to construct sequences of non-uniform lattices with systole tending to infinity, all containing a fixed submanifold group. This structural decoupling of topology and closed geodesic length is not possible in rank one settings (Hillen, 21 Mar 2024).

3. Cohomology, Torsion, and Cycle Complexity

Arithmetic manifolds exhibit deep connections between their (co)homological invariants, spectral theory, and arithmetic.

• Exponential Torsion Growth: For compact arithmetic hyperbolic 3-manifolds $X$ and local systems $M_{2k}$ defined by even symmetric powers of the standard representation of SL(2,ℂ), the order of torsion in the second cohomology has exponential growth:

$$\lim_{k \to \infty} \frac{\log |H_2(X, M_{2k})|}{k^2} = \mathrm{Vol}(X)$$

In contrast, $|H_1|$ and $|H_3|$ torsion grows much more slowly (Marshall et al., 2011).

• Cycle Complexity: There is a conjecture that in arithmetic hyperbolic 3-manifolds, $H_2(M,\mathbb{Z})$ admits a basis such that each class can be represented by a surface of genus at most $V^C$ (with $V$ the manifold volume). This conjecture is supported by both theoretical arithmetic methods (base change, automorphic cycles) and comprehensive numerical evidence (Bergeron et al., 2014).
• Cheeger–Müller Theorem and Analytic Torsion: The growth rate of torsion in homology is related to analytic torsion and regulators, governed by the Cheeger–Müller theorem. For example,

$$\frac{\log |H_1(M,\mathbb{Z})_{\mathrm{tors}}|}{\mathrm{Vol}(M)} \to \frac{1}{6\pi}$$

for congruence towers of arithmetic hyperbolic 3-manifolds (Bergeron et al., 2014).

4. Spectral Invariants and Rigidity Phenomena

Arithmetic manifolds are characterized by distinctive patterns in their spectral invariants, leading to rigidity properties and connections with automorphic forms.

• Limit Multiplicity and Local Convergence: Sequences of torsion-free arithmetic congruence lattices in rank-one groups (e.g., PSL(2,ℝ) or PSL(2,ℂ)) satisfy quantitative limit multiplicity properties: the spectral measures converge, and the volume of the thin part of the manifold is sublinear in total volume. As a consequence, these manifolds admit triangulations or presentations of their fundamental groups whose complexity is bounded linearly in volume (Fraczyk, 2016).
• Length Spectrum and Commensurability: The full length spectrum of an arithmetic hyperbolic surface determines its commensurability class, but non-commensurable manifolds can share large portions of their spectra. The number of pairwise non-commensurable arithmetic hyperbolic surfaces containing a fixed set of geodesic lengths can grow “doubly exponentially” in the cardinality of the set (Linowitz, 2017).
• Spectral Rigidity Conjectures: The presence of arithmetic progressions in the length spectrum motivates the conjecture that such spectral patterns characterize arithmeticity and that the set of negatively curved metrics with these properties is discrete or finite (Lafont et al., 2014).

5. Topology: Cusp Structures, Embeddings, and Subgroup Separability

The topology of arithmetic manifolds is governed by subtle arithmetic data and interacts with group-theoretic properties in essential ways.

• Cusp Types and Flat Manifolds: The set of flat manifolds that can arise as cusp cross-sections is determined by arithmetic constraints: a flat manifold $B$ appears as a cusp cross-section in an arithmetic commensurability class defined by a quadratic form $q$ if and only if the holonomy representation of $B$ can be conjugated into the orthogonal group O$(q)$ over $\mathbb{Q}$. For manifolds with odd-order holonomy and $b_1(B)\geq 3$, appearance is universal across classes, whereas for smaller first Betti number there are explicit numerical obstructions (McCoy et al., 14 Oct 2024).
• Totally Geodesic Embeddings: All arithmetic hyperbolic $n$-manifolds of simplest type can be geodesically embedded into higher-dimensional arithmetic hyperbolic manifolds (possibly after passing to the universal mod 2 Abelian cover in the odd-dimensional case) by extension of the underlying quadratic form (Kolpakov et al., 2017).
• NonLERFness in High Dimensions: Fundamental groups of arithmetic hyperbolic manifolds above dimension 3 (and certain compact 3-manifolds with mixed geometry) are not LERF (locally extended residually finite). This stands in stark contrast to the LERF property in finite volume hyperbolic 3-manifolds. The key mechanism is the presence of totally geodesic submanifolds and amalgamations along subgroups (Sun, 2016, Sun, 2017).
• Effective Residual Properties: Explicit effective bounds on residual finiteness growth (i.e., on the index needed to separate non-trivial elements) are available for certain arithmetic hyperbolic 3-manifold groups, via embeddings into higher-dimensional right-angled reflection groups (Deblois et al., 2018).

6. Analogues in Arithmetic Topology and Arithmetic Differential Geometry

Arithmetic manifolds inspire analogues of geometric and topological theory for number fields and arithmetic objects:

• Arithmetic Differential Geometry: The spectrum of the integers, $\mathrm{Spec}\,\mathbb{Z}$, equipped with data such as symmetric/antisymmetric matrices and “Fermat quotient” operators (p-derivations), can be considered as an “arithmetic manifold.” Curvature is introduced via commutators of Frobenius lifts and interpreted as an intrinsic geometric structure—Spec$\;\mathbb{Z}$ can be “curved” in this sense (Barrett et al., 2015).
• Arithmetic TQFT and Dijkgraaf–Witten Theory: One can assign to the spectrum of the integers, or spectra of number fields, arithmetic analogues of Chern-Simons functionals, quantum Hilbert spaces, and partition functions, mimicking topological quantum field theories for 3-manifolds. These constructions operate on local Galois data (primes as analogues of circles or links) and are organized by arithmetic gauge groups and cocycles (Hirano et al., 2021).
• Non-Diophantine Arithmetic and Effective Geometry: Changing the arithmetic structure (e.g., making a non-standard choice of addition and multiplication via a bijection $f$) can impose different global geometric or physical properties on a model space (e.g., rendering a flat Minkowski spacetime or an expanding universe in cosmology) (Czachor, 2016).

7. Open Questions, Flexibility, and Future Directions

Arithmetic manifolds are a source of ongoing questions and conjectures:

• What are the optimal constants in systole–volume and kissing number–volume inequalities across different types of locally symmetric spaces (Murillo, 31 Aug 2025)?
• Can one find sharp obstructions to the appearance of certain flat cusp cross-sections beyond current invariants, and classify the possible “spectra” of cusps within commensurability classes (McCoy et al., 14 Oct 2024)?
• Are unexpectedly large kissing numbers characteristic of arithmeticity or other group-theoretic properties?
• What are the full implications of exponential torsion growth for quantum invariants or automorphic forms, and how do torsion classes interact with Galois representations in non-cohomological settings (Marshall et al., 2011, Bergeron et al., 2014)?
• Do similar arithmeticity-dominated spectral patterns (like arithmetic progressions in the length spectrum) emerge in other symmetric spaces or in possibly non-arithmetic settings (Lafont et al., 2014, Linowitz, 2017)?

Research continues at the interface of geometric topology, arithmetic groups, spectral geometry, and number theory, with arithmetic manifolds providing foundational examples for most rigidity phenomena, for growth and asymptotic results, and for the most advanced computational and analytic techniques linking geometry and arithmetic.