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BIRS Workshop Lecture Videos

Amenability and definability Krupinski, Krzysztof


I will discuss some aspects of my recent paper (still in preparation) with Udi Hrushovski and Anand Pillay. Most of the main results are of the form "a version of amenability implies a version of G-compactness". Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some extension results that we obtain for measure-like functions (which we call means and pre-means). One of the main results is the following. Assume $G$ is a 0-definable group normalized by $G(M)$ (where $M$ is a model), $H$ is a 0-type-definable subgroup, and $N$ is the normal subgroup of $G$ generated by $H$. Then, if the type space $S_{G/H}(M)$ carries a $G(M)$-invariant, Borel probability measure, then $G^{00}_M \leq NG^{000}_M$. We also obtain a similar result which answers various questions from my earlier paper with Anand Pillay, e.g. if G is an amenable topological group, then the (classical) Bohr compactification of $G$ coincides with a certain "weak Bohr compactification". In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{top} = G^{000}_{top}$. Another main result says that each amenable theory is G-compact. In particular, we introduce the notion of an amenable theory in several equivalent ways, e.g. by saying that for each type $p(\bar x) \in S(\emptyset)$, the space $S_p(\C):={q \in S(\C): p \subset q}$ carries an $Aut(\C)$-invariant, Borel probability measure (where $\C$ is a monster model). There are several other developments in this paper, but during my talk, I will focus on one of the above.

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