Definably topological dynamics of p-adic algebraic groups

Abstract

We study the p-adic algebraic groups G from the definable topological-dynamical point of view. We consider the case where M is an arbitrary p-adic closed field and G an algebraic group over Qp admitting an Iwasawa decompostion G=KB, where K is open and definably compact over Qp, and B is a Borel subgroup of G over Qp. Our main result is an explicit description of the minimal subflow and Ellis group of the universal definable G(M)-flow SG(Mext). We prove that the Ellis group of SG(Mext) is isomorphic to the Ellis group of SB(Mext), which is B/B0.As applications, we conclude that the Ellis groups corresponding to GL(n,M) and SL(n,M) are isomorphic to (Zˆ×Zp⁎)n and (Zˆ×Zp⁎)n−1 respectively, generalizing the main result of Penazzi, Pillay, and Yao in [21].