Related papers: A non-homogeneous orbit of a diagonal subgroup

Let \Gamma be a lattice in G=SL(n,R) and X=G/S a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish uniform distribution of orbits of \Gamma in X…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More…

Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact set. As a…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give…

Let G:=SO(n,1)^\circ and \Gamma be a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(\Gamma \ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and…

Given a lattice \Gamma in a locally compact group G and a closed subgroup H of G, one has a natural action of \Gamma on the homogeneous space V=H\G. For an increasing family of finite subsets {\Gamma_T: T>0}, a dense orbit v\Gamma, v\in V,…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in R^d for d > 2 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit…

We investigate analogues of some of the classical results in homogeneous dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma<G$ a discrete subgroup. For a large class of…

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$…

In the present article we study the following problem. Let G be a linear algebraic group over Q, $\Gamma$ be an arithmetic lattice and H be an observable Q-subgroup. There is a H-invariant measure $\mu_H$ supported on the closed submanifold…

We give a simple proof about the topological rigidity of closures of certain sparse unipotent orbits in $G/\Gamma$ where $G=\prod_{i=1}^k\operatorname{SL}_2(\mathbb R)$ and $\Gamma$ is an irreducible lattice in $G$.

Let $\Gamma$ be a non-uniform lattice in $\operatorname{PSL}(2,\mathbb R)$. In this note, we show that there exists a constant $\gamma_0>0$ such that for any $0<\gamma<\gamma_0$, any one-parametrer unipotent subgroup $\{u(t)\}_{t\in\mathbb…

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in…

In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed…

We study the distribution of orbits of a lattice $\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolume one rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and our main result is the limiting…

This paper is about topological rigidity of diagonal group actions on the homogeneous $\SL_4\big(\F(\!(t^{-1})\!)\big)/\SL_4(\F[t])$ where $\F$ is a finite field of characteristic $3$. We show that there is a non-closed relatively compact…