Some model theory and topological dynamics of p-adic algebraic groups

2017/04/25 by Davide Penazzi, Anand Pillay, Penazzi, Davide +3
Mathematics · #03C45 #03C60 #22E35 #37B05 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · doi:10.48550/arxiv.1704.07764

openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields. We consider the additive and multiplicative groups of Q_p and Z_p, the group of upper triangular invertible 2× 2 matrices, SL(2,Z_p), and, our main focus, SL(2,Q_p). In all cases we identify f-generic types (when they exist), minimal subflows, and idempotents. Among the main results is that the ``Ellis group" of SL(2,Q_p) is the profinite completion of Z, yielding a counterexample to Newelski's conjecture with new features: G = G⁰⁰ = G⁰⁰⁰ but the Ellis group is infinite. A final section deals with the action of SL(2,Q_p) on the type-space of the projective line over Q_p.

Citations

• External definability and groups in NIP theories

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