Related papers: Arc-transitive digraphs with quasiprimitive local …

Let $G$ be a group of collineations of a finite thick generalised quadrangle $\Gamma$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $\Gamma$, and transitively on the lines of $\Gamma$. We show that the primitive…

Let $\Gamma$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $\Gamma$ on $\R^d$ is strongly irreducible and that $\Gamma$ contains a proximal and expanding element. We describe contraction properties of the…

Let $A$ be a Lebesgue measure space. We interpret measures on $A\times A\times R_+$ as 'maps' from $A$ to $A$, which spread $A$ along itself; their Radon-Nikodym derivatives also are spread. We discuss basic properties of the semigroup of…

We consider vertex-primitive digraphs having two vertices with almost equal neighbourhoods (that is, the set of vertices that are neighbours of one but not the other is small). We prove a structural result about such digraphs and then apply…

The relative fixity of a digraph $\Gamma$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\Gamma$ and the number of vertices of $\Gamma$. We characterize the vertex-primitive digraphs…

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $\Gamma(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus…

A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the index two subgroup…

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with…

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and…

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…

The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation…

The investigation of s-arc-transitivity of digraphs can be dated back to 1989 when the third author showed that s can be arbitrarily large if the action on vertices is imprimitive. However, the situation is completely different when the…

A non-complete graph $\Gamma$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set of $\Gamma$, and for any vertex $u$ of $\Gamma$, the stabilizer $G_u$ is…

We show that stationary characters on irreducible lattices $\Gamma < G$ of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation…

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the…

In this paper, we study the primitive actions of almost simple exceptional groups of Lie type on \(s\)-arc-transitive digraphs. Our motivation is the following question posed by Giudici and Xia: Is there an upper bound on $s$ for finite…

The commuting graph ${\Gamma(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all…

In this paper, we study almost finiteness and almost finiteness in measure of non-free actions. Let $\alpha:G\curvearrowright X$ be a minimal action of a locally finite-by-virtually $\mathbb{Z}$ group $G$ on the Cantor set $X$. We prove…

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs…

We define and develop the notion of a discretisable quasi-action. It is shown that a cobounded quasi-action on a proper non-elementary hyperbolic space $X$ not fixing a point of $\partial X$ is quasi-conjugate to an isometric action on…