Related papers: Infinite transitivity on universal torsors

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. If $M$ is connected and $\dim(M)\geq 2$, the group of diffeomorphisms of $M$, that are isotopic to the identity through a compactly supported isotopy, acts $n$-transitively…

A counterexample to the conjecture that the automorphisms of an arbitrary Arveson system act transitively on its normalized units.

We describe maximal commutative unipotent subgroups of the automorphism group $\mathrm{Aut}(X)$ of an irreducible affine variety $X$. Further we show that a group isomorphism $\mathrm{Aut}(X) \to \mathrm{Aut}(Y)$ maps unipotent elements to…

We generalize and complete some of Maxim's recent results on Alexander invariants of a polynomial transversal to the hyperplane at infinity. Roughly speaking, and surprisingly, such a polynomial behaves both topologically and algebraically…

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…

Using methods of p-adic analysis we give a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. More precisely, we show that any finitely generated torsion subgroup…

We prove that for many pairs $H_1, H_2$ of root subgroups of the automorphism group $\text{Aut}(\mathbb{C}^2)$ the diagonal action of the group generated by $H_1, H_2$ on $(\mathbb{C}^2)^m$ has an open orbit for any positive integer $m$.…

A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the index two subgroup…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in…

In this work, we present several new findings regarding the concepts of orbit-transitivity, strict orbit-transitivity, $\omega$-transitivity, and $\mu$-open-set transitivity for self-maps on generalized topological spaces. Let $(X,\mu)$…

An action of a group $G$ on a set $X$ is said to be quasi-n-transitive if the diagonal action of $G$ on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act…

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to…

We construct examples of normal affine varieties X of dimension greater than or equal to 4 such that the group of special automorphisms SAut(X) acts on X with an open orbit O and the complement X\O has codimension one.

We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi…

We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented…

We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex…

An algebraic variety $X$ is called a homogeneous variety if the automorphism group $\mathrm{Aut}(X)$ acts on $X$ transitively, and a homogeneous space if there exists a transitive action of an algebraic group on $X$. We prove a criterion of…

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each…

Let $Y\to X$ be a finite normal cover of a wedge of $n\geq 3$ circles. We prove that for any $v\neq 0\in H_1(Y;\mathbb{Q})$ there exists a lift $\widetilde{F}$ to $Y$ of a homotopy equivalence $F:X\to X$ so that the set of iterates…