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For $G$ an algebraic group definable over a model of $operatorname{ACVF}$, or more generally a definable subgroup of an algebraic group, we study the stable completion $widehat{G}$ of $G$, as introduced by Loeser and the second author. For $G$ connected and stably dominated, assuming $G$ commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable $G$-equivariant strong deformation retraction of $widehat{G}$ onto the generic type of $G$. For $G=S$ a semiabelian variety, we construct a pro-definable $S$-equivariant strong deformation retraction of $widehat{S}$ onto a definable group which is internal to the value group. We show that, in case $S$ is defined over a complete valued field $K$ with value group a subgroup of $mathbb{R}$, this map descends to an $S(K)$-equivariant strong deformation retraction of the Berkovich analytification $S^{mathrm{an}}$ of $S$ onto a piecewise linear group, namely onto the skeleton of $S^{mathrm{an}}$. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of $S$. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group $G$ is a directed union of $infty$-definable subgroups which all stabilize a generically stable Keisler measure on $G$.
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
For certain theories of existentially closed topological differential fields, we show that there is a strong relationship between $mathcal Lcup{D}$-definable sets and their $mathcal L$-reducts, where $mathcal L$ is a relational expansion of the field
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version
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 exam