Related papers: Distality of Certain Actions on $p$-adic Spheres

The norm closure of the algebra generated by the set $\{n\mapsto {\lambda}^{n^k}:$ $\lambda\in{\mathbb {T}}$ and $k\in{\mathbb{N}}\}$ of functions on $({\mathbb {Z}}, +)$ was studied in \cite{S} (and was named as the Weyl algebra). In this…

Let $G$ be a Lie group acting properly on a smooth manifold $M$. If $M/G$ is connected, then we exhibit some simple and basic constructions for proper actions. In particular, we prove that the reduction principle in compact transformation…

We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if…

Let LUC$(S)$ be the space of left uniformly continuous functions on a semitopological semigroup $S$. Suppose that $S$ is right reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fr\'echet space. Let $\tau$ be…

Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When…

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E \to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category…

For the semi simple and deployed Lie algebra $\mathfrak g=\mathfrak{sl}(n, \R)$, we give an explicit construction of an overalgebra $\mathfrak g^+=\mathfrak g\rtimes V$ of $\mathfrak g$, where $V$ is a finite dimensional vector space. In…

Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb…

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is…

This paper studies nilpotent orbits in complex simple Lie algebras from the viewpoint of strongly visible actions in the sense of T. Kobayashi. We prove that the action of a maximal compact group consisting of inner automorphisms on a…

We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and…

Let $K$ be a locally compact field of characteristic 0. Let $G$ be a linear algebraic group defined over $K$, acting algebraically on an algebraic variety $V$. We prove that the action of $G(K)$ (the group of $K$-rational points of $G$) on…

We prove that action of a semigroup T on compact metric space X by continuous selfmaps is strongly proximal if and only if T action on P(X), the space of probability measures on $X$ with weak topology, is strongly proximal. As a consequence…

We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs),…

We prove the spectral gap property for random walks on the product of two non-locally isomorphic analytic real or p-adic compact groups with simple Lie algebras, under the necessary condition that the marginals posses a spectral gap.…

We study the action of a real-reductive group $G=K\exp(\lie{p})$ on real-analytic submanifold $X$ of a K\"ahler manifold $Z$. We suppose that the action of $G$ extends holomorphically to an action of the complexified group $G^\mbb{C}$ such…

Conformal/anticonformal actions of the quasi-abelian group $QA_{n}$ of order $2^n$, for $n\geq 4$, on closed Riemann surfaces, pseudo-real Riemann surfaces and closed Klein surfaces are considered. We obtain several consequences, such as…

Let $\overline{\mathfrak{S}}_\infty$ denote the set of all bijections of natural numbers. Consider the action of $\overline{\mathfrak{S}}_\infty$ on a measure space $\left( X,\mathfrak{M},\mu \right)$, where $\mu$ is…

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL (2, Q), SL (2, Q) and SL (2, Z) groups. In…

A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist…