Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSuperstable groups acting on trees
205
0
0.0
(
0
)
Authors
Abderezak Ould Houcine
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.
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 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 $H_c^0(X, {mathcal S})$ is an irreducible representation of $G$, then $H_c^0(X, {mathcal S})$ arises by induction from a vertex or edge stabilizing subgroup. If $G$ is a reductive group over a nonarchimedean local field $F$, then Schneider and Stuhler realize every irreducible supercuspidal representation of $G(F)$ in the degree-zero cohomology of a $G(F)$-equivariant sheaf on its reduced Bruhat-Tits building $X$. When the derived subgroup of $G$ has relative rank one, $X$ is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.
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 field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $omega_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
