### $(*)$-groups and pseudo-bad groups.

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The purpose of this paper is to prove the existence of a free subgroup of the group of all affine transformations on the plane with determinant 1 such that the action of the subgroup is locally commutative.

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

We show that for no infinite group $G$ the class of abelian-by-$G$ groups is elementary, but, at least when $G$ is an infinite elementary abelian $p$-group (with $p$ prime), the class of groups admitting a normal abelian subgroup whose quotient group is elementarily equivalent to $G$ is elementary.

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case). We also study minimal groups with a (partial) order, and give a complete classification of...

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the...

Let $G$ be an uncountable universal locally finite group. We study subgroups $H<G$ such that for every $g\in G$, $|H:H\cap {H}^{g}|<|H|$.

We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon’s completion ${G}^{\mathbb{Z}\left[t\right]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of ${G}^{\mathbb{Z}\left[t\right]}$ containing $G$ is universally equivalent to $G$. Since finitely generated...