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

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 x 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 (Z) over cap x 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 SL(2, Q(p)) on the type space of the projective line over Q(p).