Related papers: Dixmier Groups and Borel Subgroups

We study the actions of a Lie group $G$ by birationally extendible automorphisms on a domain $D\subset C^n$. For a large class of such domains defined by polynomial inequalities, all automorphisms are of this type. In the cases 1) $G$ has…

Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\mathcal{P}$ must admit an odd…

We study topological quivers $Q$ admitting a free and proper action by a locally compact group $G$ together with their associated $C^*$-algebras. On the topological side, we provide a complete classification of topological quivers which…

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system \lambda. The twisted groupoid C*-algebra C*(E;G,\lambda) is a quotient of the C*-algebra…

We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of…

In this paper, we introduce the notion of regular action of any abelian subgroup G of $Diff^{1}(C^n) on C^n (i.e. the closure of every orbit of G in some open set is a topological sub-manifold of C^n). We prove that if G fixes 0 and…

Let $\Aut(G)$ denote the group of (bi-)continuous automorphisms %and $\Out(G)$ the outer automorphism group of a non-Archimedean Polish group~$G$. We show that for any such $G$ with an invariant countable basis of open subgroups, the group…

Let S be the Stone space of a complete, non-atomic Boolean algebra. Let G be a countably infinite group of homeomorphisms of S. Let the action of G on S have a free dense orbit. Then we prove that, on a generic subset of S, the orbit…

Let $G$ be a subgroup of $S_{n}$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for…

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the…

A. Borel proved that, if a finite group $F$ acts effectively and continuously on a closed aspherical manifold $M$ with centerless fundamental group $\pi_1(M)$, then a natural homomorphism $\psi$ from $F$ to the outer automorphism group…

We prove a few basic facts about the space of bi-invariant (or left-invariant) total order relations on a torsion-free, nonabelian, nilpotent group G. For instance, we show that the space of bi-invariant orders has no isolated points (so it…

Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield…

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 topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann…

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy…

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction of a quotient group G/N of a discrete group G to a C*-coaction of G itself on an induced C*-algebra. We…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex…

Let $G=G_{n}=GL(n)$ be the $n\times n$ complex general linear group and embed $G_{n-1}=GL(n-1)$ in the top left hand corner of $G$. The standard Borel subgroup of upper triangular matrices $B_{n-1}$ of $G_{n-1}$ acts on the flag variety of…

Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The…