arXiv subjects
Publications and source records attributed to C. R. E. Raja.
We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.
Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$.
We consider linear groups and Lie groups over a non-Archimedean local field $\mathbb F$ for which the power map $x\mapsto x^k$ has a dense image or it is surjective. We prove that the group of $\mathbb F$-points of such algebraic groups is a compact extension of unipotent groups with the order of the compact group being relatively prime to $k$. This in particular shows that the power map is surjective for all $k$ is possible only when the group is unipotent or trivial depending on whether the characteristic of $\mathbb F$ is zero or positive. Similar results are proved for Lie groups via the adjoint representation. To a large extent, these results are extended to linear groups over local fields and global fields.
A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.
We study automorphisms $α$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $α^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $α$ on $G/N$ for an $α$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_α$ consisting of all $x$ in $G$ such that $α^n(x) \to e$ as $n$ tends to infinity. If $α$ is expansive, then $W := U_αU_{α^{-1}}$ is an open identity neighbourhood in $G$. We give examples where $W$ fails to be a subgroup. However, $W$ is a nilpotent open subgroup whenever $G$ is a closed subgroup of a general linear group over the $p$-adic numbers. Further results are devoted to the divisible and torsion parts of $U_α$, and to the so-called "nub" $U_0$ of an expansive automorphism $α$ (the intersection of the closures of $U_α$ and $U_{α^{-1}}$).
We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup $K$ such that $G/K$ is distal and the conjugacy action of $G$ on $K$ is ergodic; moreover, if $G$ itself is (pointwise) distal then $G$ is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.
We provide a sufficient condition for a $p$-adic Lie group to be of type $R$ when its adjoint image is of type $R$.
We consider the following convolution equation or equivalently stochastic difference equation $$\lam _k = μ_k*ϕ(\lam _{k-1}), k \in \Z \eqno (1) $$ for a given bi-sequence $(μ_k)$ of probability measures on $\R ^d$ and a linear map $ϕ$ on $\R ^d$. We study the solutions of equation (1) by realizing the process $(μ_k)$ as a measure on $(\R ^d)^\Z$ and rewriting the stochastic difference equation as $\lam = μ*τ(\lam )$-any such measure $\lam$ on $(\R ^d)^\Z$ is known as $τ$-decomposable measure with co-factor $μ$-where $τ$ is a suitable weighted shift operator on $(\R ^d)^\Z$. This enables one to study the solutions of (1) in the settings of $τ$-decomposable measures. A solution $(\lam _k)$ of (1) will be called a fundamental solution if any solution of (1) can be written as $\lam _k*ϕ^k(ρ)$ for some probability measure $ρ$ on $\R ^d$. Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one-one correspondence between fundamental solutions of (1) and strongly $τ$-decomposable measures on $(\R ^d)^\Z$ with co-factor $μ$. We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent $\ell _p$-paths.
Let $G$ be a locally compact group and $E$ be a $G$-space. An irreducible probability measure $μ$ on $G$ is said to have Liouville property on $E$ if $G$-invariant functions on $E$ are the only continuous bounded functions on $E$ that satisfy the mean value property with respect to $μ$. We first prove that the random walk induced by $μ$ on $E$ is transient outside a closed set and on the closed set $μ$ has Liouville. We mainly consider actions on vector spaces and projective spaces. We show that measures on $GL(V)$ that are supported inside a ball of radius less than $a<1$ have Liouville property on $V$. We also prove that measures on $GL(\R ^2)$ have Liouville property on the projective line. We next exhibit subgroups of $GL(V)$ so that irreducible measures on such subgroups have Liouville on the projective space $\mP (V)$ of $V$. We also prove irreducible measures on $SL(V)$ have Liouville property on $\mP (\cSL (V))$ where $\cSL (V)$ is the Lie algebra of $SL(V)$
We consider strong relative property $(T)$ for pairs $(\Ga, G)$ where $\Ga$ acts on $G$. If $N$ is a connected Lie group and $\Ga$ is a group of automorphisms of $N$, we choose a finite index subgroup $\Ga ^0$ of $\Ga$ and obtain that $(\Ga, [\Ga ^0, N])$ has strong relative property $(T)$ provided Zariski-closure of $\Ga$ has no compact factor of positive dimension. We apply this to obtain the following: $G$ is a connected Lie group with solvable radical $R$ and a semisimple Levi subgroup $S$. If $S_{nc}$ denotes the product of noncompact simple factors of $S$ and $S_T$ denotes the product of simple factors in $S_{nc}$ that have property $(T)$, then we show that $(\Ga, R)$ has strong relative property $(T)$ for a Zariski-dense closed subgroup of $S_{nc}$ if and only if $R=[S_{nc},R]$. The case when $N$ is a vector group is discussed separately and some interesting results are proved. We also considered actions on solenoids $K$ and proved that if $\Ga$ acts on a solenoid $K$, then $(\Ga, K)$ has strong relative property $(T)$ under certain conditions on $\Ga$. For actions on solenoids we provided some alternatives in terms of amenability and strong relative property $(T)$. We also provide some applications to the spectral gap of $π(μ)=\int π(g) dμ(g)$ where $π$ is a certain unitary representation and $μ$ is a probability measure.
We consider the stochastic difference equation $$η_k = ξ_k ϕ(η_{k-1}), ~~~~ k \in \Z $$ on a locally compact group $G$ where $ξ_k$ are given $G$-valued random variables, $η_k$ are unknown $G$-valued random variables and $ϕ$ is an automorphism of $G$. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $ξ_k$ have a common law $μ$ and prove that if $G$ is a pointwise distal group and $ϕ$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $μ$ is supported on $Kz= zϕ(K)$ for any $z$ in the support of $μ$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.
In this note, we give an original convergence result for products of independent random elements of motion group. Then we consider dynamic random walks which are inhomogeneous Markov chains whose transition probability of each step is, in some sense, time dependent. We show, briefly, how Central Limit theorem and Local Limit theorems can be derived from the classical case and provide new results when the rotations are mutually commuting. To the best of our knowledge, this work represents the first investigation of dynamic random walks on the motion group.
We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular we show that a closed subgroup of a product of finitely many linear groups over local fields supports a recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces. The structure of closed subgroups of linear groups over local fields and the properties of group actions with respect to stationary measures play an important role in the proofs.
Let $K$ be a compact metrizable group and $\Ga$ be a finitely generated group of commuting automorphisms of $K$. We show that ergodicity of $\Ga$ implies $\Ga$ contains ergodic automorphisms if center of the action, $Z(\Ga) = \{\ap \in {\rm Aut}(K) \mid \ap {\rm commutes with elements of \rm} \Ga \}$ has DCC. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which in particular, retrieves Theorems of Berend \cite{Be} and Schmidt \cite{Sc1} proved in this context.
A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $μ$ on $G$, either $\sup_{x\in G} μ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that $μ^nx^{-n} \ra ω_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.
Let $K$ be a compact metrizable group and $\Ga$ be a group of automorphisms of $K$. We first show that each $\ap \in \Ga$ is distal on $K$ implies $\Ga$ itself is distal on $K$, a local to global correspondence provided $\Ga$ is a generalized $\ov{FC}$-group or $K$ is a connected finite-dimensional group. We show that $\Ga$ contains an ergodic automorphism when $\Ga$ is nilpotent and ergodic on a connected finite-dimensional compact abelian group $K$.
We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contraction subgroup of every inner automorphism of $G$ is trivial; (v) $G$ is a SIN group. We also show that for every probability measure $μ$ on a totally disconnected compactly generated locally compact second countable group of polynomial growth, the Poisson boundary is a homogeneous space of $G$, and that it is a compact homogeneous space when the support of $μ$ generates $G$.
A sequence $a_n\da 0$ forms a rate of random mixing for a unitary system $(G,μ, π, {\cal H})$ if for any $u, v\in {\cal H}$ $$\limsup {|<π(g_n^ω)u, v>|\over a_n} < \infty$$ a.e. $ω$ in the probability space $(G^{\mathbb N}, μ^{\mathbb N})$ of the random walk induced by $μ$. We study the class of locally compact groups none of whose representation has any rate of random mixing and prove that this class contains Moore groups and certain solvable groups which includes the group of affine transformations on a local field.