Related papers: Groups Acting Generically Multiply Transitively on…

We continue our study of the perfect kernel of the space of transitive actions of Baumslag-Solitar groups by investigating high transitivity. We show that actions of finite phenotype are never highly transitive, except when the phenotype is…

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and…

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group $G$ acts…

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0.…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime $p$. Recently, a particular action of the group $\mathrm{GL}_2(\mathbb F_q)$ on irreducible polynomials in $\mathbb F_q[x]$ has been introduced and…

The paper is devoted to generalizations of actions of topological groups on manifolds. Instead of a topological group, we consider a local topological group generalizing the notion of a~germ or a~neighborhood in a topological group. The…

A group is said to be self-similar provided it admits a faithful state-closed representation on some regular $m$-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level…

Let X be an algebraic variety covered by open charts isomorphic to the affine space and q: X' \to X be the universal torsor over X. We prove that the automorphism group of the quasiaffine variety X' acts on X' infinitely transitively. Also…

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\partial\colon K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating…

The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M.…

For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ itself is an Abelian group with respect to…

Let $L(G)$ denote the space of integer-valued length functions on a countable group $G$ endowed with the topology of pointwise convergence. Assuming that $G$ does not satisfy any non-trivial mixed identity, we prove that a generic (in the…

For a complex reductive group G acting linearly on a complex affine space V with respect to a character, we show two stratifications of V associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal…

We define the decomposition property for partial actions of discrete groups on $C^*$-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions…

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$,…

We show that every irreducible, simply connected curve on a toric affine surface X over the field of complex numbers is an orbit closure of a multiplicative group action on X. It follows that up to the action of the automorphism group…

Given a connected reductive algebraic group $G$ and a Borel subgroup $B \subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their…

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$…

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying…

We give the first examples of (non-amenable group) amenable actions on stably finite simple C*-algebras. More precisely, we give such actions for any countable group in an explicit way. The main ingredients of our construction are the full…