Related papers: Algebraic and context-free subsets of subgroups

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all…

We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $\ell^2$-invisibility,…

Given a locally finite leafless tree $ T $, various algebraic groups over local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are isomorphic to a…

We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions)…

A 'prehomogeneous vector space' is a rational representation $\rho:G\to\mathrm{GL}(V)$ of a connected complex linear algebraic group $G$ that has a Zariski open orbit $\Omega\subset V$. Mikio Sato showed that the hypersurface components of…

Let $$1 \to H \to G \to Q \to 1$$ be an exact sequence where $H= \pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide…

We describe the structure of the quotient $\mathfrak{G}/\mathfrak{H}$ of a formal supergroup $\mathfrak{G}$ by its formal sub-supergroup $\mathfrak{H}$. This is a consequence which arises as a continuation of the authors' work (partly with…

We explore connected affine algebraic groups $G$, which enjoy the following finiteness property $\rm (F)$: for every algebraic action of $G$, the closure of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main results.…

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units…

We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In…

For a finite group $H$ and connected topological spaces $X$ and $Y$ such that $X$ is endowed with a free left $H$-action $\tau$, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the…

In this paper we study hypergraphs definable in an algebraically closed field. Our goal is to show, in the spirit of the so-called transference principles in extremal combinatorics, that if a given algebraic hypergraph is "dense" in a…

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $\omega$-categorical rings…

We show that the biflatness - in the sense of A. Ya. Helemskii - of the Fourier algebra $A(G)$ of a locally compact group $G$ forces $G$ to either have an abelian subgroup of finite index or to be non-amenable without containing $F_2$, the…

Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap…

The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be context-free, then the group is called context-free. Finitely…

This paper studies a subset of the free semi-group $F_k$ with letters in a field $k$ which has some interesting arithmetic and combinatorial properties.

Let G be a group and f be an endomorphism of G. A subgroup H of G is called f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups that are f-inert for all inner automorphisms of G are widely known and studied in…

A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by…

The free product of an arbitrary pair of finite hyperfinite von Neumann algebras is examined, and the result is determined to be the direct sum of a finite dimensional algebra and an interpolated free group factor $L(\freeF_r)$. The finite…