Linear Groups Over Real Closed Fields

• Linear groups over real closed fields are groups of invertible matrices defined by polynomial equations, characterized by ordered field properties and semialgebraic structure.
• They generalize classical Lie theory by transferring decompositions, homological invariants, and building structures via the Tarski–Seidenberg transfer principle.
• Positivity concepts and valuation techniques extend representation theory, linking advanced K-theory and geometric constructions with applications in higher Teichmüller and geometric group theory.

A linear group over a real closed field is a group of invertible matrices with entries in a real closed field, typically studied as the group of F-rational points of a semisimple linear algebraic group defined over F. The algebraic, representation-theoretic, and geometric structures realized by such groups are central to the extension of classical Lie theory, higher Teichmüller theory, and geometric group theory to non-Archimedean and semialgebraic settings. Recent research has established precise generalizations of homological, structural, and representation-theoretic results—previously known only for real Lie groups—to the context of real closed fields, using the model-theoretic Tarski–Seidenberg transfer principle, valuation-theoretic analysis, and positivity concepts adapted from higher Teichmüller theory (Coronado, 2021, Flamm, 2022, Appenzeller, 12 Jan 2026, Appenzeller, 6 Jan 2026, Appenzeller, 2024, Flamm et al., 8 Jan 2026).

1. Real Closed Fields as Base Fields

A real closed field F is an ordered field in which every positive element is a square and every odd-degree polynomial has a root. Real closed fields are uniquely characterized as ordered fields whose algebraic closure is a quadratic extension: F(i) is algebraically closed, with i² = –1. Examples include ℝ, the field of real algebraic numbers, Puiseux series over ℝ, and various non-Archimedean real closed fields (Flamm et al., 8 Jan 2026).

These fields admit a model-theoretic transfer principle: any first-order statement in the language of ordered fields that holds over ℝ holds over any real closed F, and vice versa. This enables the semialgebraic transfer of structural theorems and geometric constructions from the classical real case to general real closed fields, even when F is non-Archimedean or highly transcendental (Appenzeller, 12 Jan 2026, Appenzeller, 2024).

2. Structural Theory of Linear Groups Over Real Closed Fields

Let G be a connected semisimple linear algebraic group defined over a subfield K ⊂ ℝ, and F a real closed field containing K. The group G(F) is an affine algebraic F-group: it is the set of F-points of a Zariski-closed subgroup of GL_n defined by polynomial equations with coefficients in K. Subgroups defined by polynomial equalities and inequalities (semialgebraic subsets) are termed linear semialgebraic F-groups (Appenzeller, 12 Jan 2026, Appenzeller, 2024).

Analogues of classical structures from Lie group theory persist. The maximal F-split tori, the root datum, Cartan and Iwasawa decompositions, Bruhat decompositions, and parabolic subgroup structures transfer verbatim to G(F) via first-order definability. For instance:

• Iwasawa decomposition: G(F) = K_F A_F U_F, where K_F is the "maximal compact" semialgebraic subgroup, A_F is the maximal F-split torus (identity component), and U_F is the unipotent radical (Appenzeller, 12 Jan 2026).
• Cartan decomposition: G(F) = K_F A_F K_F, with uniqueness of the A_F part up to the spherical Weyl group action.
• Bruhat decomposition: G(F) = ⨆_{w∈W_s} B_F w B_F, for B_F = M_F A_F U_F where M_F is the centralizer of A_F in K_F (Appenzeller, 2024).

Semialgebraic versions of the Baker–Campbell–Hausdorff formula, the Jacobson–Morozov lemma, and Kostant's convexity theorem hold in G(F), guaranteeing rich structural and representation theory (Appenzeller, 12 Jan 2026, Appenzeller, 2024).

3. Homological Invariants and K-Theory Connections

The third integral homology H₃(SL_n(F), ℤ), and its relation to higher K-theory, exhibits unique features over real closed fields. For R a real closed field:

• H₃(SL₂(R), ℤ) ≅ K₃^{ind}(R), the indecomposable K₃ group, via identification with the refined Bloch group, utilizing scissors-congruence relations and the collapse of 4-torsion due to the triviality of R×/(R×)² ≅ {±1} (Coronado, 2021).
• For n ≥ 3, H₃(SL_n(R), ℤ) ≅ H₃(SL₂(R), ℤ) ⊕ K₃^{M}(R)^0, where K₃^{M}(R)^0 denotes the Milnor K₃ subgroup generated by {x,y,z} for x, y, z > 0. This direct sum decomposition is uniquely 2-divisible and isolates the "indecomposable" from the "fully Milnor" components.
• The refined Bloch group approach clarifies that analytic or measure-theoretic arguments are unnecessary—the full computation proceeds via algebraic K-theory and the order structure of R (Coronado, 2021).

4. Positivity, Hitchin, and Θ-Positive Representations

The positivity theory, originating in the study of higher Teichmüller spaces, admits a semialgebraic extension. Let G = PSL(n, F).

• A representation ρ: π₁(S) → PSL(n, F) is F-positive if there exists a π₁(S)-equivariant (possibly discontinuous) map ξ from fixed points at infinity into Flag(F^n), the flag variety, satisfying Fock–Goncharov positivity for all cyclically ordered tuples. This includes F-valued triple ratios and double ratios as positivity tests (Flamm, 2022).
• The space of F-positive representations forms a semi-algebraically connected component in the moduli space of reductive representations π₁(S) → PSL(n, F).
• For Δ- (all simple roots) or Θ-positive representations, the positivity property is phrased via nonempty Fuchsian-invariant subsets D ⊂ ∂∞ and boundary maps sending cyclically ordered tuples to positive tuples in F_Θ, with positivity defined through semialgebraic connected components of unipotent radicals (Flamm et al., 8 Jan 2026).
• When F is non-Archimedean, flag varieties become totally disconnected, yet positivity structures and the corresponding dynamical and geometric properties persist at the semialgebraic level.

These representations extend the concept of Anosov representations, with a precise correspondence between positivity, weak Θ-proximality, and maximal equivariant positive boundary maps (Flamm, 2022, Flamm et al., 8 Jan 2026).

5. Symmetric Spaces and Affine Λ-Buildings

Symmetric space structures for G(F) are realized as sets of F-symmetric positive-definite matrices (for SL_n), with the action of G(F) via conjugation (Appenzeller, 2024). For F equipped with a valuation v: F× → Λ (Λ a divisible ordered abelian group):

• One constructs an affine Λ-building B, with apartments modeled on the coroot lattice L ⊗_ℤ Λ ≅ Λ^r, and distance function d(x,y) = –v(N_F(δ_F(x,y))) for the multiplicative root-norm N_F and Cartan projection δ_F.
• The building structure encodes the combinatorics of the Weyl group, apartments, root valuations, and decomposition theorems: Bruhat, Cartan, Iwasawa, and Levi decompositions for G(F) (Appenzeller, 6 Jan 2026).
• The spherical building at infinity and local residue buildings at base points (identified with flag varieties over the residue field) are realized through group-theoretic stabilizers and root group data.
• In the classical case G = GLₙ or SLₙ, the Λ-building recovers the matrix- and valuation-theoretic structure of the Bruhat–Tits A_{n−1}-building, fully encoded in semialgebraic terms (Appenzeller, 6 Jan 2026, Appenzeller, 2024).

6. Model-Theoretic and Valuation-Theoretic Transfer

All major structural and geometric theorems for real Lie groups, symmetric spaces, buildings, and representation varieties extend to real closed base fields through the Tarski–Seidenberg transfer principle. This tool allows first-order definable properties—such as group decompositions, convexity, and representation positivity—to be formulated and proven over arbitrary real closed fields, regardless of Archimedean or non-Archimedean character (Appenzeller, 12 Jan 2026, Appenzeller, 2024, Appenzeller, 6 Jan 2026).

Valuation structures enrich the geometry and permit the definition of affine buildings with value group metrics, opening new connections between algebraic group theory, non-Archimedean geometry, and moduli of linear representations.

7. Summary Table: Key Structural Features

Phenomenon	ℝ (classical)	Real Closed F (general)
Group Decompositions (Iwasawa, Cartan, etc.)	Differentiable; analytic	Semialgebraic, transferred
Homology H₃(SL₂(F), ℤ)	Bloch group methods	Refined Bloch, K₃^{ind}
Buildings	Symmetric space, Tits	Λ-building, valuation-based
Θ-positivity, Hitchin/Anosov	Differential topology	Semialgebraic, positivity
Model theory transfer	Not needed	Essential, Tarski–Seidenberg

All major results are established via semialgebraic or first-order definability and transfer, enabling a unified framework for the structure, homology, and representation theory of linear groups over arbitrary real closed fields (Coronado, 2021, Flamm, 2022, Appenzeller, 12 Jan 2026, Appenzeller, 2024, Appenzeller, 6 Jan 2026, Flamm et al., 8 Jan 2026).