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

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 Qp in the language of fields. We consider the additive and multiplicative groups of Qp and Zp, the group of upper triangular invertible 2 × 2 matrices, SL(2, Zp), and, our main focus, SL(2, Qp).
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, Qp ) is Zˆ, 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, Qp) on the type-space of the projective line over Qp.