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Register a new userSome model theory and topological dynamics of p-adic algebraic groups
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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 2times 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 Newelskis 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.
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Let ${mathfrak C}$ be a monster model of an arbitrary theory $T$, $bar alpha$ any tuple of bounded length of elements of ${mathfrak C}$, and $bar c$ an enumeration of all elements of ${mathfrak C}$. By $S_{bar alpha}({mathfrak C})$ denote the compact space of all complete types over ${mathfrak C}$ extending $tp(bar alpha/emptyset)$, and $S_{bar c}({mathfrak C})$ is defined analogously. Then $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$ are naturally $Aut({mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{bar c}({mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{bar c}({mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{bar alpha}({mathfrak C})$ in place of $S_{bar c}({mathfrak C})$.
In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore the different aspects of this connection.
For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Delta$-definable $G$-flows for any fixed set $Delta$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{Delta,M}$ and ${G^*}^{000}_{Delta,M}$, and show that $G^*/{G^*}^{00}_{Delta,M}$ is the $Delta$-definable Bohr compactification of $G$. We also note that some deeper arguments from the topological dynamics in the category of externally definable $G$-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the $Delta$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $Delta$-definable generalized Bohr compactification to the $Delta$-definable Bohr compactification of $G$. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.
This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups definable in stable, simple and NIP theories, definable compactifications of groups, definable Galois theory (including differential Galois theory), connections with topological dynamics, model theory of the free group.
We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorv{c}evi{c} theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todorv{c}evi{c}, Moore, Ngyuen Van Th{e}, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.
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