We study definably amenable groups in N IP theories, focusing on the problem raised in [10] of whether weak generic types coincide with almost periodic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We give fairly definitive results in the o-minimal context, including a counterexample.
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 × 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 ) iŝ 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 .
We study the flow (G(Q p ), S G (Q p )) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable fgeneric types, equivalently whether the union of minimal subflows of a suitable type space is closed. We will give a description of of f -generic types of trigonalizable algebraic groups, and prove that every f -generic type is almost periodic.
In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.