# Questions tagged [group-actions]

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487
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### Identification of Maharam extension

All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...

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### Definition of free group action on a scheme

This may be an elementary question, but I didn't know what it means for a group action on a scheme to be free from the textbooks I read. I guess that at least for schemes locally of finite ...

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### How much a general a theory of matrices equivalence under group actions we have?

Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$.
My question is: Do we have some theory about the ...

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### Liouville-Arnold and fibration relative to a convex polytope

Liouville-Arnold's theorem indicates that given a Hamiltonian torus action on a manifold and a set of $n$ functions $F$ from the manifold to $\mathbb{R}^n$ defining an integrable system, the pre image ...

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### What is the quotient of the unit sphere by the bilateral shift on infinite-dimensional separable real Hilbert space?

Let H denote the real Hilbert space 𝓁2(ℤ) with its usual inner product.
If {en | n ∈ ℤ} denotes its standard orthonormal basis, define the unitary mapping W : H → H via W(en) = en+1, extended ...

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### Why $d\mu (q)\delta (k,q)$ is $G$-invariant?

Let $G$ be a Lie group acting transitively on a smooth manifold $M$ endowed with a quasi-invariant measure $\mu$ (then there exists Radon-Nikodym derivative $\rho_f$ for every $f\in G$). For $k\in M$,...

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### Isbell duality for monoids and groups

Isbell Duality
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Sets}{\mathsf{Sets}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\B}{\...

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### Equivariant disk theorem in dimension 2

All groups I'll consider are finite.
An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...

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219
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### Hopf algebras actions

Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...

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### What is a cogroup and what are coactions?

What is a cogroup and what are coactions?
A very nice way to think about a group action on an object $X$ is as a group homomorphism from $G$ to $\operatorname{Aut}(X)$. Is there something similar for ...

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### Orbits/affine spaces in GAP

Another GAP-related question.
I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...

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116
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### Expressing the union of principal orbits as a disjoint union of global slices for proper group actions

Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...

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### Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...

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### Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...

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163
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### References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994).
I would like to ask for references on semigroup actions on ...

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### Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group

Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$.
Next, let $\mathcal{S}(\...

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### Action of algebraic group in cohomology of equivariant algebraic vector bundle

Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...

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### Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions
$$N\rightarrow G\rightarrow Q$$
are classified by a group homomorphism
$$\rho: Q\rightarrow \operatorname{Out}(N)$$
subject to a constraint ...

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218
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### If a discrete and faithful representation of a surface group has proximal values, does the attracting points map have a continuous extension?

For some context, I'm studying the paper Anosov Representations and Proper Actions [GGKW].
$G$ denotes a non-compact real reductive Lie group of rank greater than $1$, $\Gamma$ denotes the fundamental ...

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### Almost simple groups and their involutions without CFSG

Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...

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### Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...

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### Name for a product of actions / dynamical systems

Suppose $G \curvearrowright X, H \curvearrowright Y$ are group (or monoid) actions, or dynamical systems. Then $X \times Y$ is a $G \times H$-system of the same type in the obvious way by $(g, h) \...

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### Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...

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117
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### Smooth action on cotangent space of the plane

Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by
$$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$
acts via ...

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### Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on

For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...

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172
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### Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...

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154
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### Examples of discrete quantum group actions on commutative von Neumann algebras

Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...

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### A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?

Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...

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### Action of braid groups on regular trees

Question:
Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...

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118
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### Nonamenable p.m.p. action on a standard probability space

Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, ...

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149
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### An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere

Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$?
A counterexample to a more general question can be ...

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47
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### Holomorphic cyclic action on smooth toric manifold extends to C^* action?

Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?

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107
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### Integral mean value property

Let $V$ be the space of all continuous functions $f$ on the real line with $f(x)=\frac12\big(f(x-1)+f(x+1)\big)$.
It contains the space of periodic functions. The latter equals the space of ...

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### Bialynicki-Birula decomposition for $\mathbb{G}_m$-actions on projective schemes

The classical BB-decomposition works for non-singular projective varieties. Here I want to consider projective schemes, in particular when the scheme is not reduced.
Let $\Bbbk=\mathbb{C}$. Let $X$ be ...

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### Additional symmetries in Theta-like function

cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function
Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows
$$ \...

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### S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...

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### Lifting paths along group quotients relative to a base

Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly ...

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### Groups acting on categories produce 2-cocycles

$\DeclareMathOperator\Hom{Hom}\newcommand\id{\mathrm{id}}\DeclareMathOperator\Aut{Aut}$Let $\mathcal{C}$ be a category (such that each hom sets are $\mathbb{C}$ linear spaces) and $G$ be a group. We ...

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441
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### dichotomy in hyperbolic groups

Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...

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### Semi-direct products and associated graded Lie algebras

Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...

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### How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?

Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...

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### Examples of amenable, Hausdorff, locally compact, second countable groups which are not discrete, not compact, and not abelian

I'm working on a problem that involves an amenable group acting on some set by bijections. Initially, I assumed the group was discrete and the set was countable, however I realized that the arguments ...

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422
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### Group action on a condensed set and its orbit space

Let $X$ be a condensed set, and let $G$ be a (discrete) group. Suppose we have an action $G$ on $X$, which is a group morphism $a:G \rightarrow \mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ is the group ...

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### Lifting a group action to a Banach bundle

I have been searching the literature for results on lifting a group action from the base space of a Banach bundle (Important note: NOT Banach VECTOR bundle). The setting I am interested in weaker than ...

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### Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...

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279
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### A possible invariant associated to a compact group

Let $G$ be a compact topological group with normalized Haar measure $\mu$.
Is there an effective isometric action of $G$ on some $\mathbb{R}^n$ such that the following map would be a non-irreducible ...

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### Do we have an equivariant version of integrability theorem of flat connections?

I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1:
Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...

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### Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?

Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers.
Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....

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115
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### Unitary in adjointable operators associated with equivariant Hilbert module

Consider the following fragment from the article "Tannaka–Krein duality for compact quantum
homogeneous spaces. I. General theory" by De Commer and Yamashita:
How exactly is $\mathcal{E}\...

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### Different invariants of group actions from isomorphic subgroups

Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...