Related papers: The existential transversal property: a generaliza…

In this paper we introduce the definition of $(k,l)$-universal transversal property, which is a refinement of the definition of $k$-universal transversal property, which in turn is a refinement of the classic definition of $k$-homogeneity…

Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group $G\leq \sym$ is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$.…

We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if…

Given an action of a group $G$ by automorphisms on an infinite relational structure $\mathcal{M}$, we say that the action is structurally sharply $k$-transitive if, for any two $k$-tuples $\bar{a}, \bar{b} \in M^k$ of distinct elements such…

We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an…

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a…

For a connected quasi-split reductive algebraic group $G$ over a field $k$, which is either a finite field or a non-archimedean local field, $\theta$ an involutive automorphism of $G$ over $k$, let $K =G^\theta$. Let $K^1=[K^0,K^0]$, the…

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to…

This paper discusses `geometric property (T)'. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of `expansion property': in particular…

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth…

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of…

A subset $S$ of a transitive permutation group $G \leq \mathrm{Sym}(n)$ is said to be an intersecting set if, for every $g_{1},g_{2}\in S$, there is an $i \in [n]$ such that $g_{1}(i)=g_{2}(i)$. The stabilizer of a point in $[n]$ and its…

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible…

Let $ G $ be a connected reductive algebraic group over a field $ k $. We study the group of semilinear automorphisms Aut($ G\to $Spec $k$) consisting of algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the exact…

Let $k_0$ be a field of characteristic $p>0$ and $k=k_0(t)$, where $t$ is transcendental over $k_0$. We give an example of a smooth connected unipotent $k$-group $G$ such that $G(F)/R$ is non-commutative for some finite separable field…

We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type…

In this paper we study the category $\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\mathcal{K}$ of weights we define a subquotient $\mathcal{O}_{[\mathcal{K}]}$…

Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 Steinberg proved that if $G$ is simply connected, then in $G$ there exists a closed irreducible cross-section of the set of closures of…

The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. Let $G$ be such a group. We prove…

The $k$-gonal models of random groups are defined as the quotients of free groups on $n$ generators by cyclically reduced words of length $k$. As $k$ tends to infinity, this model approaches the Gromov density model. In this paper we show…