Related papers: Character rigidity of simple algebraic groups

Let G be the set of k-points of a connected semisimple algebraic group of k-rank one over a nonarchimedean local field k. We describe all finitely generated torsion-free discrete subgroups of G\times G acting properly discontinuously and…

Let k be an algebraically closed field of positive characteristic and G a simple algebraic group defined over k. Under the assumption that the characteristic is a good prime for G, we determine a maximal G-stable subvariety U' of the…

Let G be a connected linear algebraic group over a field k. We say that G is toric-friendly if for any field extension K/k and any maximal K-torus T in G the group G(K) has only one orbit in (G/T)(K). Our main result is a classification of…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

Let $G$ be an absolutely almost simple simply connected algebraic group defined over a number field $K$, and let $M/K$ be the minimal Galois extension over which $G$ becomes an inner form of a split group. Assume that $G$ satisfies the…

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of…

Let $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms…

We prove: (1) The group of multipliers of similitudes of a 12-dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_E is…

Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$…

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let S be the Steinberg representation of G(K). Let t be be a very regular semisimple element of G(K). In this paper we give a…

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type…

For every algebraically closed field $\boldsymbol k$ of characteristic different from $2$, we prove the following: (1) Generic finite dimensional (not necessarily associative) $\boldsymbol k$-algebras of a fixed dimension, considered up to…

The celebrated Borel--Tits theorem provides a classification of abstract isomorphisms between (simple) isotropic groups over fields, showing that such isomorphisms arise from field isomorphisms and group-scheme isomorphisms. In this work,…

The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial…

We prove that every ergodic amenable action of an algebraic group over a local field of characteristic zero is induced from an ergodic action of an amenable subgroup.

Let $G$ be a commutative connected algebraic group over a number field $K$, let $A$ be a finitely generated and torsion-free subgroup of $G(K)$ of rank $r>0$ and, for $n>1$, let $K(n^{-1}A)$ be the smallest extension of $K$ inside an…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite…

Let $G$ be a connected reductive group over a perfect field $k$ acting on an algebraic variety $X$ and let $P$ be a minimal parabolic subgroup of $G$. For $k$-spherical $G$-varieties we prove finiteness result for $P$-orbits that contain…

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…