Related papers: Confined subgroups and high transitivity

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

A transitive smooth action of a connected Lie group G on a manifold M is called almost primitive (resp. primitive) if G doesn't contain any proper subgroup (resp. any proper normal subgroup) whose induced action on M is transitive as well.…

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of…

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this…

A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number e>0 such that for any open set U we can find g in G such that g.U has diameter greater than e. We prove that if a…

For a finite group $A$ with normal subgroup $G$, a subgroup $U$ of $G$ is an $A$-prime-power-covering subgroup if $U$ meets every $A$-conjugacy-class of elements of $G$ of prime power order. It is conjectured that $|G:U|$ is bounded by some…

Let G be a compact group. Let (X,G) be a standard Borel G-measure space. We show that the group action on (X, G) is transitive if and only if it is ergodic. Using this result, we show that every irreducible covariant representation of a…

Let $\text{Homeo}_{+}(\mathbb{S}^1)$ denote the group of orientation preserving homeomorphisms of the circle $\mathbb{S}^1$. A subgroup $G$ of $\text{Homeo}_{+}(\mathbb{S}^1)$ is tightly transitive if it is topologically transitive and no…

A directed graph $G=(V,E)$ is {\it strongly pseudo transitive} if there is a partition $\{A,E-A\}$ of $E$ so that graphs $G_1=(V,A)$ and $G_2=(V,E-A)$ are transitive, and additionally, if $ab\in A$ and $bc\in E $ implies that $ac\in E$. A…

We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such…

Two subgroups H and K are 4-permutable in G if <H,K> = HKHK, and H is strong 4-quasinormal in G if H is 4-permutable with every subgroups K of G. A finite group G is called Sq4T-group if strong 4-quasinormality is transitive relation among…

We study hereditary properties of the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We consider mainly the case of amalgamated free products, and we show in particular that the double of…

Topological groupoids admit various types of morphisms. We push these notions to the level of continuous groupoid actions to obtain various types of groupoid action morphisms. Some dynamical properties and their relation to these morphisms…

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the…

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.)…

Let $X$ be an algebraic variety such that the group $\text{Aut}(X)$ acts on $X$ transitively. We define the transitivity degree of $X$ as a maximal number $m$ such that the action of $\text{Aut}(X)$ on $X$ is $m$-transitive. If the action…

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let (C,G) be a "big action", i.e. a pair (C,G) where G is a p-subgroup of the k-automorphism group of C such…

A group $G$ is invariably generated (IG) if there is a subset $S \subseteq G$ such that for every subset $S' \subseteq G$, obtained from $S$ by replacing each element with a conjugate, $S'$ generates $G$. $G$ is finitely invariably…

We show that any isometric action of a residually finite group admits approximate local finite models. As a consequence, if $G$ is residually finite, every isometric $G$-action embeds isometrically into a metric ultraproduct of finite…

Let $G$ be a finite group acting transitively on a set $\Omega$. We study what it means for this action to be {\it quasirandom}, thereby generalizing Gowers' study of quasirandomness in groups. We connect this notion of quasirandomness to…