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We study superstable groups acting on trees. We prove that an action of an $omega$-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not $omega$-stable. It is also shown that if $G$ is a superstable group acting nontrivially on a $Lambda$-tree, where $Lambda=mathbb Z$ or $Lambda=mathbb R$, and if $G$ is either $alpha$-connected and $Lambda=mathbb Z$, or if the action is irreducible, then $G$ interprets a simple group having a nontrivial action on a $Lambda$-tree. In particular if $G$ is superstable and splits as $G=G_1*_AG_2$, with the index of $A$ in $G_1$ different from 2, then $G$ interprets a simple superstable non $omega$-stable group. We will deal with minimal superstable groups of finite Lascar rank acting nontrivially on $Lambda$-trees, where $Lambda=mathbb Z$ or $Lambda=mathbb R$. We show that such groups $G$ have definable subgroups $H_1 lhd H_2 lhd G$, $H_2$ is of finite index in $G$, such that if $H_1$ is not nilpotent-by-finite then any action of $H_1$ on a $Lambda$-tree is trivial, and $H_2/H_1$ is either soluble or simple and acts nontrivially on a $Lambda$-tree. We are interested particularly in the case where $H_2/H_1$ is simple and we show that $H_2/H_1$ has some properties similar to those of bad groups.
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups
In this paper, the notion of proper proximality (introduced in [BIP18]) is studied for various families of groups that act on trees. We show that if a group acts non-elementarily by isometries on a tree such that for any two edges, the intersection o
We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
If $G$ is a group acting on a tree $X$, and ${mathcal S}$ is a $G$-equivariant sheaf of vector spaces on $X$, then its compactly-supported cohomology is a representation of $G$. Under a finiteness hypothesis, we prove that if $H_c^0(X, {mathcal S})$
Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular