Lie Group-Theoretic Conjecture

• Lie group-theoretic conjectures are propositions that explore the structure, representation, and spectral properties of Lie groups, linking algebraic, analytic, and geometric insights.
• They employ reduction frameworks and equivalences, such as Margulis's and Mathieu's conjectures, to transform complex non-abelian problems into more tractable abelian settings.
• These conjectures drive interdisciplinary research by establishing deep correspondences between number theory, topology, and differential geometry, inspiring new proofs and methodologies.

A Lie group-theoretic conjecture is, in its broadest sense, a conjectural statement whose formulation and significance intrinsically involve the structure, representation, or geometry of Lie groups and their associated algebraic, analytic, or topological objects. Such conjectures are central to numerous mathematical disciplines, including representation theory, number theory, geometry, and algebraic topology. This article surveys some of the most influential Lie group-theoretic conjectures, their interrelations with arithmetic, geometry, and cohomology, and illustrates how the field has crystallized around several core universality principles and reduction frameworks.

1. Fundamental Definitions and Examples

A Lie group is a group $G$ that is also a smooth manifold such that the group operations are smooth. Lie group-theoretic conjectures typically make nontrivial claims about properties of $G$, its discrete subgroups, measure-theoretic invariants, cohomology, representations, or their interaction with algebraic structures such as polynomials or number fields.

Prominent examples include:

• Margulis's Uniform Discreteness Conjecture: For connected semisimple real Lie groups $G$ with $\mathrm{rank}_{\,\mathbb{R}}(G)\ge2$, Margulis conjectured that there exists a neighborhood $U$ of the identity in $G$ such that, for every irreducible cocompact lattice $\Gamma < G$, $U \cap \Gamma$ consists only of torsion elements. This statement controls the local behavior of lattices in higher-rank Lie groups and encodes strong rigidity properties (Pham et al., 2020).
• Mathieu Conjecture: For any compact connected Lie group $K$, the vanishing of all moments $\int_K f^n\,dk = 0$ for a finite-type function $f$ implies, for any other finite-type $h$, that $\int_K f^n h \, dk = 0$ for all but finitely many $n$. This conjecture provides a group-theoretic formulation closely linked to the celebrated Jacobian Conjecture (Zwart, 20 Nov 2025).

2. Arithmetic Reductions and Logical Equivalences

One of the most remarkable recent developments is the proof that apparently distant conjectures in Lie group theory and number theory can be logically equivalent. A primary instance is the equivalence between Margulis's uniform discreteness conjecture for cocompact lattices in all higher-rank semisimple Lie groups and a weak form of Lehmer's conjecture from Diophantine approximation.

Main Result

Let $G$ be as above. The weak form of Lehmer's conjecture posits: For each integer $s \geq 1$, there exists $\varepsilon(s) > 0$ such that for every monic $P \in \mathbb{Z}[X]$ with at most $s$ roots outside the unit circle, $M(P) = 1$ or $M(P) > 1+\varepsilon(s)$, where $M(P)$ is the Mahler measure. Pham and Thilmany proved:

• Theorem (Pham–Thilmany): For each fixed $s \ge 1$, Margulis's conjecture holds for every group in a specific family $T_F^{(s)}$ of almost-simple isotropic $\mathbb{R}$–groups if and only if the weak Lehmer conjecture at level $s$ holds. In particular, the full Margulis conjecture is equivalent to the full weak Lehmer conjecture (Pham et al., 2020).

This equivalence is established by matching spectral gaps in the eigenvalue distribution (Lie theory) with Mahler measure gaps of polynomials (number theory), via the eigenvalues of the adjoint action of lattice elements in arithmetic groups.

3. Reduction Frameworks and Abelianization

For several deep Lie group-theoretic conjectures, a central methodological advance is reduction to statements about commutative or abelian settings. This is especially manifest for the Mathieu Conjecture and related problems, as shown in recent advances:

• Explicit Haar Integral Formula and Reduction: Müger and Tuset developed an explicit formula for the Haar integral on compact connected Lie groups via Poisson geometry and Bruhat cell decompositions, using Kostant harmonic forms. They constructed a linear reduction that maps finite-type functions on $K$ to explicit polynomial-Laurent polynomial objects on $[0,1]^N \times T^M$, with integration against a specific weight (Müger et al., 15 Oct 2024).
• Abelian Moment Conjectures: The original Mathieu conjecture can thereby be reformulated as a statement about integrals of Laurent polynomials with monomial weights over cubes times tori. In particular, it reduces to the question of whether the convex hull of the exponent spectrum of such a polynomial contains the origin, which is a convex-geometric property independent of any Lie group (Müger et al., 15 Oct 2024).

Similarly, Zwart's sharp reduction of the Mathieu conjecture for $SU(N)$, $Sp(N)$, and $G_2$ witnesses that the moment vanishing problem can be fully transferred to polydisks times tori, decoupled from non-abelian Lie-theoretic phenomena (Zwart, 2 Apr 2025).

4. Representation Theory, Cohomology, and Model Categories

Lie group-theoretic conjectures frequently manifest as compatibility statements between algebraic or topological invariants associated to $G$ and those associated to some combinatorial or discrete abstraction.

• Friedlander–Milnor Conjectures: These posit that mod-$p$ cohomology of classifying spaces agrees for a Lie group and its discrete version: $$H^*(BG^{\delta};\mathbb{F}_p) \cong H^*(BG;\mathbb{F}_p)$$ for all $p$ (Amrani, 2015). The proof exploits model-category techniques for topological groups, pushouts, and rational localization, showing equivalence results for stable linear groups as well.
• Mathieu Subspace and Representation-Theoretic Reductions: For the standard representation $V$ of $SL(N)$, the Mathieu conjecture can be stated in terms of the vanishing of projections onto highest-weight submodules in tensor algebras, controlled by explicit $SL(N)$ module theory, thus recasting a functional-analytic vanishing into purely representation-theoretic constraints (Zwart, 20 Nov 2025).

5. Related Conjectures and Connections

Lie group-theoretic conjectures often interact with, or are equivalent to, foundational conjectures in geometry, topology, and analysis:

• Farrell–Jones Conjecture: Asserts that algebraic $K$- and $L$-theory of group rings $A[G]$ is determined by equivariant homology for certain discrete groups $G$. This conjecture is known to hold for cocompact lattices in virtually connected Lie groups, using the Farrell–Hsiang method, flow-geometry constructions, and induction over subgroup hierarchies (Bartels et al., 2011).
• Schmid–Vilonen Conjecture: Predicts that signatures of natural Hermitian forms on Harish–Chandra modules of a real reductive group are governed by underlying Hodge and weight filtrations, connecting D-module theory on flag varieties with unitarity and representation theory (Schmid et al., 2015).
• Larsen–Shalev Conjecture: Concerns word maps with coefficients in simple compact Lie groups: any non-constant word map $w_G:G^d \to G$ is surjective if $Z(G) = \{e\}$, established for several root systems using ping-pong arguments in representations spaces (Larsen et al., 2022).

6. Broader Impact and Open Problems

Lie group-theoretic conjectures establish deep correspondences across number theory, group theory, differential geometry, and mathematical logic:

• Universality: The equivalence between uniform discreteness for higher-rank lattices and Lehmer-type gaps demonstrates how spectral and Diophantine rigidity are sometimes two sides of the same phenomenon. A plausible implication is that further arithmetic differential geometry can be encoded in modularity or periodicity statements for lattice spectra.
• Arithmetic Algebraization: The full abelianization of integration problems opens a pathway for attacking classical problems, such as the Jacobian conjecture, via convexity and moment theory, harnessing tools from real algebraic geometry instead of Lie theory.
• Outstanding Questions: The full (non-abelian) Mathieu conjecture remains open, associated convex-hull conjectures are unsettled in multiple variables, Margulis's conjecture relies on the resolution of generalizations of Lehmer's conjecture, and generalizations of Schmid–Vilonen to arbitrary real reductive groups and their functorial filtrations are incomplete.
• Extensions: Current research seeks to extend these conjectures to general S-arithmetic groups, to K-theory with twisted coefficients, and to broader classes of topological groups and their cohomology.

7. Selected Table of Lie Group-Theoretic Conjectures

Conjecture	Core Statement (paraphrased)	Principal Reference
Margulis Uniform Discreteness	Cocompact lattices in higher rank are uniformly discrete	(Pham et al., 2020)
Weak Lehmer	Mahler measures with $s$ outside roots bounded from $1$	(Pham et al., 2020)
Mathieu (vanishing moments)	Vanishing powers of finite-type $f$ force further vanishings	(Zwart, 20 Nov 2025, Zwart, 2 Apr 2025)
Friedlander–Milnor	Top. group cohomology equals discrete group cohomology	(Amrani, 2015)
Farrell–Jones	Assembly isomorphism in algebraic K/L-theory	(Bartels et al., 2011)
Schmid–Vilonen	Hodge-theoretic realization of unitarity	(Schmid et al., 2015)
Larsen–Shalev	Non-constant word maps are surjective	(Larsen et al., 2022)

Lie group-theoretic conjectures function as organizing principles that unify group-theoretic, geometric, and arithmetic structures. Their paper continues to stimulate advances in logic, topology, arithmetic geometry, and representation theory, often by exhibiting surprising equivalences and reduction mechanisms across disparate mathematical domains.