Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the topological dynamics of automorphism groups; a model-theoretic perspective
176
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
For a group $G$ first order definable in a structure $M$, we continue the study of the definable topological dynamics of $G$. The special case when all subsets of $G$ are definable in the given structure $M$ is simply the usual topological dynamics of the discrete group $G$; in particular, in this case, the words externally definable and definable can be removed in the results described below. Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant $G^{*}/(G^{*})^{000}_{M}$ of $G$, which appears to be new in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalized Bohr compactification of $G$; [externally definable] strong amenability. Among other things, we essentially prove: (i) The new invariant $G^{*}/(G^{*})^{000}_{M}$ lies in between the externally definable generalized Bohr compactification and the definable Bohr compactification, and these all coincide when $G$ is definably strongly amenable and all types in $S_G(M)$ are definable, (ii) the kernel of the surjective homomorphism from $G^*/(G^*)^{000}_M$ to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup, and (iii) when $Th(M)$ is NIP, then $G$ is [externally] definably amenable iff it is externally definably strongly amenable. In the situation when all types in $S_G(M)$ are definable, one can just work with the definable (instead of externally definable) objects in the above results.
We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of generic used by Akin, Glasner, and Weiss is distinct from the notion of generic encountered in Fraisse theory.
We will review the main results concerning the automorphism groups of saturated structures which were obtained during the two last decades. The main themes are: the small index property in the countable and uncountable cases; the possibility of recovering a structure or a significant part of it from its automorphism group; the subgroup of strong automorphisms.
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $omega$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors. We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
