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 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\times 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 = G^{00} = G^{000} 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.
Cite
@article{arxiv.1704.07764,
title = {Some model theory and topological dynamics of p-adic algebraic groups},
author = {Davide Penazzi and Anand Pillay and Ningyuan Yao},
journal= {arXiv preprint arXiv:1704.07764},
year = {2019}
}
Comments
The new version is 24 pages. Compared to version 2, there are more detailed proofs, and there is a correction in the description of the Ellis group of SL(2,Q_p) acting on its type space