Related papers: Confined subgroups and high transitivity

We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d, 1, n)$ (the group of $d$-colored permutations) and $G(d, d, n)$.

Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is…

Let $G$ be a group with socle a simple group of Lie type defined over the finite field with $q$ elements where $q$ is a power of the prime $p$. Suppose that $G$ acts transitively upon the lines of a linear space $\mathcal{S}$. We show that…

We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of…

Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix…

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally…

We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we…

Masures are generalizations of Bruhat--Tits buildings and the main examples are associated with almost split Kac--Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure…

Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let…

An action trace is a function naturally associated to a probability measure preserving action of a group on a standard probability space. For countable amenable groups, we characterise stability in permutations using action traces. We…

Given groupoids $G$ and $H$ and a $(G,H)$-equivalence $X$ we may form the transformation groupoid $G\ltimes X\rtimes H$. Given a separable groupoid dynamical system $(A,G\ltimes X\rtimes H,\omega)$ we may restrict $\omega$ to an action of…

We construct a collection of numerical invariants for approximately transitive (AT) actions (of $\Z$). We use them (sometimes supplemented by other invariants to show that members of various one-parameter families of AT actions are mutually…

We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case…

Let $G$ be a permutation group, and denote with $\mu(G)$ and $b(G)$ its minimal degree and base size respectively. We show that for every $\varepsilon>0$, there exists a transitive permutation group $G$ of degree $n$ with \[ \mu(G)b(G) \geq…

The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we…

It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Cech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete…

Let K be a fine hyperbolic graph and G be a group acting on K with finite quotient. We prove that G is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups…

We prove that every acylindrically hyperbolic group admits a minimal and extremely proximal action on a compact metrizable space. If there are no nontrivial finite normal subgroups, then the action is topologically free. This answers…

We show that if G is a discrete group which does not have the Haagerup property but does have an unbounded cocycle into a C_0 representation and if G acts on a finite von Neumann algebra B such that the inclusion B \subset (B \rtimes G) has…