Related papers: R-equivalence and A^1-connectedness in anisotropic…

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive…

Seeking for a converse to a well-known theorem by Borel-Tits, we address the question whether the group of rational points G(k) of an anisotropic reductive k-group may admit a split spherical BN-pair. We show that if k is a perfect field or…

We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\exists\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily…

We show that a compact open subgroup $H$ of a simple algebraic $p$-adic group $G$ is self-similar if and only if it is isotropic.

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$

Given a simple, acyclic dimension group $G_{0}$ and countable, torsion-free, abelian group $G_{1}$, we construct a minimal, amenable, \'{e}tale equivalence relation $R$ on a Cantor set whose associated groupoid $C^{*}$-algebra, $C^{*}(R)$,…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

We prove the stable rationality of almost simple algebraic groups, the connected components of the Dynkin diagram of anisotropic kernel of which contain at most two vertices. The (stable) rationality of many isotropic almost simple groups…

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if…

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

Suppose $\rho_1, \rho_2$ are two $\ell$-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally…

We prove that a one-relator group $G$ is K\"ahler if and only if either $G$ is finite cyclic or $G$ is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus $g > 0$ with at most one cone point of order $n$: $$<…

Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when…

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X=G/H. We ask: How many connected components…

Let $G$ be a reductive group, and let $X$ be a smooth quasi-projective complex variety. We prove that any $G$-irreducible, $G$-cohomologically rigid local system on $X$ with finite order abelianization and quasi-unipotent local monodromies…

A conjecture of Morel asserts that the sheaf of A^1-connected components of a simplicial sheaf X is A^1-invariant. A conjecture of Asok-Morel asserts that A^1-connected components of smooth k-schemes coincide with their A^1-chain-connected…

We show that if G is a split semisimple algebraic group over a model complete field K, then the groups G(K) and G(K)' (the commutator group which is a ``Chevalley group'' as for example the group PSL_2(K)) are model complete as well.

Let $\mathcal{G}=\mathrm{Spec}(A)$ be a finite and flat group scheme over the ring of algebraic integers $R$ of a number field $K$ and suppose that the generic fiber of $\mathcal{G}$ is the constant group scheme over $K$ for a finite group…

Let $G$ be a $5$-group of maximal class and $\gamma_2(G) = [G, G]$ its derived group. Assume that the abelianization $G/\gamma_2(G)$ is of type $(5, 5)$ and the transfers $V_{H_1\to \gamma_2(G)}$ and $V_{H_2\to \gamma_2(G)}$ are trivial,…