Model-Theoretic Homogeneity of Affine and Hyperbolic Coxeter Groups — E8 Intelligence Research

FINDING: Affine Coxeter groups and many hyperbolic Coxeter groups are homogeneous in the model-theoretic sense, while split crystallographic groups are not. MATH: - Homogeneity condition: For any two finite tuples \(ā, b̄\) in a structure \(M\) with the same type, there exists an automorphism \(σ ∈ Aut(M)\) such that \(σ(ā) = b̄\). - Affine Coxeter groups correspond to root systems of types \(Ã_n, B̃_n, C̃_n, D̃_n, Ẽ_6, Ẽ_7, Ẽ_8, F̃_4, G̃_2\). - Hyperbolic Coxeter groups include compact and non-compact types (e.g., \([3,3,6]\), \([3,4,4]\), \([5,3,4]\)). - Split crystallographic groups: semidirect products \(Z^n W\) where \(W\) is a finite Coxeter group acting by reflections on the lattice. CONNECTION: - Affine Coxeter groups encode the symmetry of infinite lattices (e.g., root lattices \(A_n, D_n, E_8\) extended by translations). Their homogeneity re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com