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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 of their edge stabilizers is finite, then G is properly proximal. We then provide a complete classification result for proper proximality among graph products of non-trivial groups, generalizing recent work of Duchesne, Tucker-Drob and Wesolek classifying inner amenability for graph products. As a consequence of the above result we obtain the absence of Cartan subalgebras and Cartan-rigidity in properly proximal graph products of weakly amenable groups with Cowling-Haagerup constant 1.
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
We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually o
A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For groups of h
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 t
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 $H_c^0(X, {mathcal S})$