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Riddhi Shah

Publications and source records attributed to Riddhi Shah.

17 recordsLinked to original sources
arXiv2025

Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications

For a connected Lie group $G$, we study the dynamics of actions of automorphisms of $G$ on certain compact invariant subspaces of closed subgroups of $G$ in terms of distality and expansivity. We show that only the finite order automorphisms of $G$ act distally on Sub$^p_G$, the smallest compact space containing all closed one-parameter subgroups of $G$, when $G$ is any $n$-torus, $n\in\mathbb{N}$. This enables us to relate distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of closed subgroups generate by $T$ in the group Aut$(G)$, in case $G$ is not a vector group. We also extend these results to the action of subgroups of automorphisms. We show that any $n$-torus $G$, $n\geq 2$, more generally, any connected Lie group $G$ whose central torus has dimension at least 2, does not admit any automorphism which acts expansively on Sub$^p_G$. Our results generalise some results on distal actions by Shah and Yadav, and by Chatterjee and Shah, and some results on expansive actions by Prajapati and Shah.

math.DS↗
arXiv2025

Twisted conjugacy classes in Lie groups

We consider twisted conjugacy classes of continuous automorphisms $\varphi$ of a Lie group $G$. We obtain a necessary and sufficient condition on $\varphi$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when $G$ is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group $G$ that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group $G$, there exists $n\in \mathbb{N}$ such that Reidemeister number of $\varphi^n$ is infinite for every $\varphi$. We say that $G$ has topological $R_\infty$-property if the Reidemeister number of every $\varphi$ is infinite. We obtain conditions on a connected solvable Lie group under which it has topological $R_\infty$-property; which, in particular, enables us to prove that the group of invertible $n\times n$ upper triangular real matrices and its quotient group modulo its center have topological $R_\infty$-property for every $n\geq 2$. We also prove that the Walnut group also has this property. We show that ${\mathrm{SL}}(2,\mathbb{R})$ and ${\mathrm{GL}}(2,\mathbb{R})$ have topological $R_\infty$-property, and construct many examples of Lie groups with this property.

math.GR↗
arXiv2025

Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups

For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$, and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Some of our results extend those of Shah and Yadav.

math.DS↗
arXiv2025

On the almost algebraicity of groups of automorphisms of connected Lie groups

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let ${\rm Aut}(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of ${\rm GL} (\frak G)$, where $\frak G$ is the Lie algebra of $G$. It is known that when $C$ is trivial ${\rm Aut}(G)$ is almost algebraic, in the sense that it is of finite index in an algebraic subgroup of ${\rm GL}(\frak G)$, and in particular has only finitely many connected components. In this paper we analyse the situation further in this respect, with $C$ possibly nontrivial, and describe necessary and sufficient conditions for almost algebraicity to hold; the criteria are in terms of the group of restrictions of automorphisms of $G$ to $C$, and the abelian quotient Lie group $G/\overline{[G,G]}C$. For the class of Lie groups which admit a finite-dimensional representation with discrete kernel, a more specific criterion for ${\rm Aut}(G)$ to be almost algebraic is obtained, while in the general case a variety of patterns are illustrated through examples. Along the way we also study almost algebraicity of subgroups of ${\rm Aut}(G)$ fixing each point of a given torus in $G$, containing $C$, which also turns out to be of independent interest.

math.GR↗
arXiv2023

Cartan subgroups in connected locally compact groups

We define Cartan subgroups in connected locally compact groups, which extends the classical notion of Cartan subgroups in Lie groups. We prove their existence and justify our choice of the definition which differs from the one given by Chevalley on general groups. Apart from proving some properties of Cartan subgroups, we show that the Cartan subgroups of the quotient groups are precisely the images of Cartan subgroups of the ambient group. We establish the so-called `Levi' decomposition of Cartan subgroups which extends W\"ustner's decomposition theorem and our earlier results for Lie groups. We also show that the centraliser of any maximal torus of the radical is connected and its Cartan subgroups are also Cartan subgroups of the ambient group; moreover, every Cartan subgroup arises this way. We prove that Cartan subalgebras defined by Hofmann and Morris in pro-Lie algebras are the same as those corresponding to Cartan subgroups in case of pro-Lie algebras of connected locally compact groups, and that they are nilpotent. We characterise density of the image of a power map in a connected locally compact group in terms of its surjectivity on all Cartan subgroups, and show that weak exponentiality of the group is equivalent to the condition that all its Cartan subgroups are connected.

math.GR↗
arXiv2021

Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

For a locally compact metrizable group $G$, we study the action of $\rm Aut(G)$ on $\rm Sub_G$, the set of closed subgroups of $G$ endowed with the Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of the $T$-action on $\rm Sub_G$ with that of the $T$-action on $G$ under a certain condition. If $G$ is a connected Lie group, we characterise the distality of the $T$-action on $\rm Sub_G$ in terms of compactness of the closed group generated by $T$ in $\rm Aut(G)$ under certain conditions on the center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of positive dimension or $T$ is unipotent or $T$ is contained in the connected component of the identity in $\rm Aut(G)$. Moreover, we also show that a connected Lie group $G$ acts distally on $\rm Sub_G$ if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on $\rm Sub^a_G$, a subset of $\rm Sub_G$ consisting of closed abelian subgroups of $G$.

math.DS↗
arXiv2020

Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups

For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study the structure and properties of lattices $Γ$ in a connected Lie group. In particular, we show that the unique maximal solvable normal subgroup of $Γ$ is polycyclic and the corresponding quotient of $Γ$ is either finite or admits a cofinite subgroup which is a lattice in a connected semisimple Lie group with certain properties. We also show that Sub$^c_Γ$, the set of cyclic subgroups of $Γ$, is closed in Sub$_Γ$. We prove that an infinite discrete group $Γ$ which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on Sub$^c_Γ$, while only the finite order automorphisms of $Γ$ act distally on Sub$^c_Γ$. For an automorphism $T$ of a connected Lie group $G$ and a $T$-invariant lattice $Γ$ in $G$, we compare the behaviour of the actions of $T$ on Sub$_G$ and Sub$_Γ$ in terms of distality. We put certain conditions on the structure of the Lie group $G$ under which we show that $T$ acts distally on Sub$_G$ if and only if it acts distally on Sub$_Γ$. We construct counter examples to show that this does not hold in general if the conditions on the Lie group are relaxed.

math.GR↗
arXiv2020

The Structure of Cartan Subgroups in Lie Groups

We study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises Wüstner's structure theorem for the same. We show that Cartan subgroups are same as those of the centralizers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We also study the density of the images of power maps on a connected Lie group and show that the image of any $k$-th power map has dense image if its restriction to a closed normal subgroup and the corresponding map on the quotient group have dense images.

math.GR↗
arXiv2020

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\rm Sub}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. For a certain class of discrete groups $G$, we show that $T$ acts distally on ${\rm Sub}_G$ if and only if $T^n$ is the identity map for some $n\in{\mathbb N}$. As an application, we get that for a $T$-invariant lattice $Γ$ in a simply connected nilpotent Lie group $G$, $T$ acts distally on ${\rm Sub}_G$ if and only if it acts distally on ${\rm Sub}_Γ$. This also holds for any closed $T$-invariant co-compact subgroup $Γ$. For a lattice $Γ$ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on ${\rm Sub}_Γ$. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups $G$, we obtain the following characterisation: $T$ acts distally on ${\rm Sub}_G$ if and only if $T$ is contained in a compact subgroup of ${\rm Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\rm Sub}_G$. We also show that any compactly generated distal group $G$ is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups.

math.DS↗
arXiv2020

Expansive Actions of Automorphisms of Locally Compact Groups $G$ on ${\rm Sub}_G$

For a locally compact metrizable group $G$, we consider the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$, the space of all closed subgroups of $G$ endowed with the Chabauty topology. We study the structure of groups $G$ admitting automorphisms $T$ which act expansively on ${\rm Sub}_G$. We show that such a group $G$ is necessarily totally disconnected, $T$ is expansive and that the contraction groups of $T$ and $T^{-1}$ are closed and their product is open in $G$; moreover, if $G$ is compact, then $G$ is finite. We also obtain the structure of the contraction group of such $T$. For the class of groups $G$ which are finite direct products of $\mathbb{Q}_p$ for distinct primes $p$, we show that $T\in{\rm Aut}(G)$ acts expansively on ${\rm Sub}_G$ if and only if $T$ is expansive. However, any higher dimensional $p$-adic vector space $\mathbb{Q}_{p^n}$, ($n\geq 2$), does not admit any automorphism which acts expansively on ${\rm Sub}_G$.

math.DS↗
arXiv2019

Distality of Certain Actions on $p$-adic Spheres

Consider the action of $GL(n,\mathbb{Q_p})$ on the $p$-adic unit sphere $\mathcal{S}_n$ arising from the linear action on $\mathbb{Q}_p^n\setminus\{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $GL(n,\mathbb{Q}_p)$ on $\mathcal{S}_n$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on $\mathcal{S}_n$. (2) the closure of the image of $\mathfrak{S}$ in $PGL(n,\mathbb{Q}_p)$ is a compact group. On $\mathcal{S}_n$, we consider the `affine' maps $\overline{T}_a$ corresponding to $T$ in $GL(n,\mathbb{Q}_p)$ and a nonzero $a$ in $\mathbb{Q}_p^n$ satisfying $\|T^{-1}(a)\|_p<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_a$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on $\mathcal{S}_n$. The dynamics of `affine' maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

math.DS↗
arXiv2018

Dynamics of Certain Distal Actions on Spheres

Consider the action of $SL(n+1,\mathbb{R})$ on $\mathbb{S}^n$ arising as the quotient of the linear action on $\mathbb{R}^{n+1}\setminus\{0\}$. We show that for a semigroup $\mathfrak{S}$ of $SL(n+1,\mathbb{R})$, the following are equivalent: $(1)$ $\mathfrak{S}$ acts distally on the unit sphere $\mathbb{S}^n$. $(2)$ the closure of $\mathfrak{S}$ is a compact group. We also show that if $\mathfrak{S}$ is closed, the above conditions are equivalent to the condition that every cyclic subsemigroup of $\mathfrak{S}$ acts distally on $\mathbb{S}^n$. On the unit circle $\mathbb{S}^1$, we consider the `affine' actions corresponding to maps in $GL(2,\mathbb{R})$ and discuss the conditions for the existence of fixed points and periodic points, which in turn imply that these maps are not distal.

math.DS↗
arXiv2017

Relative Property (T) for Nilpotent Subgroups

We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. This and other results are a consequence of a theorem of independent interest, which states that if $H$ is a closed subgroup of a locally compact group $G$, and $A$ is a closed subgroup of the center of $H$, such that $A$ is normal in $G$, and $(G/A, H/A)$ has relative Property (T), then $(G, H^{(1)})$ has relative Property (T), where $H^{(1)}$ is the closure of the commutator subgroup of $H$. In fact, the assumption that $A$ is in the center of $H$ can be replaced with the weaker assumption that $A$ is abelian and every $H$-invariant finite measure on the unitary dual of $A$ is supported on the set of fixed points.

math.RT↗
arXiv2015

Some Properties of Distal Actions on Locally Compact Groups

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.

math.DS↗
arXiv2011

On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 1.6).

math.PR↗
arXiv2010

Orbits of Distal Actions on Locally Compact Groups

We discuss properties of orbits of (semi)group actions on locally compact groups G. In particular, we show that if a compactly generated locally compact abelian group acts distally on G then the closure of each of its orbits is a minimal closed invariant set (i.e. the action has [MOC]). We also show that for such an action distality is preserved if we go modulo any closed normal invariant subgroup and hence [MOC] is also preserved. We also show that any semigroup action on G has [MOC] if and only if the corresponding actions on a compact invariant metrizable subgroup K and on the quotient space G/K has [MOC].

math.DS↗