Some model theory and topological dynamics of $p$-adic algebraic groups
Volume 247 / 2019
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 ${\mathbb Q}_p$ in the language of fields. We consider the additive and multiplicative groups of ${\mathbb Q}_p$ and ${\mathbb Z}_p$, the group of upper triangular invertible $2\times 2$ matrices, ${\rm SL}(2,{\mathbb Z}_p)$, and our main focus, ${\rm SL}(2,{\mathbb Q}_p)$. 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 ${\rm SL}(2,{\mathbb Q}_p)$ is $\hat{\mathbb Z}\times {\mathbb Z}_p^{*}$, 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 ${\rm SL}(2,{\mathbb Q}_p)$ on the type space of the projective line over ${\mathbb Q}_p$.
