# Questions tagged [group-actions]

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465
questions

5
votes

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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 $\mathscr{B}$...

4
votes

1
answer

110
views

### 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 $M$ ...

1
vote

0
answers

105
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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 $...

4
votes

1
answer

321
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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 ...

1
vote

1
answer

72
views

### 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, ...

4
votes

1
answer

141
views

### 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 ...

0
votes

1
answer

46
views

### 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?

0
votes

1
answer

103
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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 ...

2
votes

0
answers

65
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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 ...

3
votes

0
answers

75
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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
$$ \...

6
votes

0
answers

125
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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 ...

2
votes

0
answers

49
views

### 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 ...

1
vote

0
answers

143
views

### 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 ...

8
votes

1
answer

383
views

### 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 ...

3
votes

0
answers

112
views

### 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 ...

0
votes

0
answers

51
views

### 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 ...

0
votes

0
answers

57
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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 ...

-1
votes

1
answer

159
views

### Character tables of semidirect products on Sage

I am trying to find the character table of a semidirect product of two group with Sage. If I try the following I get an error.
...

5
votes

1
answer

364
views

### 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 ...

2
votes

0
answers

32
views

### 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 ...

0
votes

0
answers

66
views

### 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}
...

1
vote

0
answers

278
views

### 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 ...

3
votes

1
answer

82
views

### 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 ...

0
votes

0
answers

151
views

### 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 $R_F$....

0
votes

0
answers

28
views

### Is there a sofic approximation for the free group on two generators satisfying $\sigma_i (a)(j)=j+1$?

Consider $\{a,b\}$ the generators of the free group.
With a sofic approximation I mean a sequence $\{\sigma_i\}_{i=1}^{\infty}$, with $\sigma_i:F_2\to \text{Sym}(d_i)$ satisfying
$\lim_{i\to\infty}\...

0
votes

0
answers

95
views

### Example of two equivariant structures on the same coherent sheaf which do not differ by a grading shift

Suppose we have a variety $X$ with a $\mathbb{C}^*$-action. If $\mathcal F$ is a $\mathbb C^*$-equivariant coherent sheaf on $X$ and $m \in \mathbb Z$, we define the grading shift ${\mathcal F}\{m\}$ ...

0
votes

0
answers

102
views

### When are all $\mathbb C^*$-equivariant structures on the structure sheaf $\mathcal O$ grading shifts of the trivial equivariant structure?

Suppose we have a variety $X$ with a $\mathbb{C}^*$-action. If $\mathcal F$ is a $\mathbb C^*$-equivariant coherent sheaf on $X$ and $m \in \mathbb Z$, we define the grading shift ${\mathcal F}\{m\}$ ...

3
votes

1
answer

103
views

### 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}\...

2
votes

0
answers

48
views

### 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 ...

11
votes

2
answers

880
views

### Not very transitive actions

Suppose $m$ is a positive integer.
I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-...

1
vote

0
answers

39
views

### What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?

$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...

2
votes

1
answer

132
views

### Generalization of $G/T \simeq G_\mathbb{C}/B$

Let $G$ be a compact Lie group and Let $G_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote ...

1
vote

0
answers

46
views

### Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions

Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...

1
vote

0
answers

94
views

### Exists $G$-equivariant embedding with faithful representation of $G$?

Let $k$ be a field of characteristic zero and $G$ a reductive group over $k$. Furthermore, let $X$ be a projective $k$-variety with a $G$-action. Then we know, for example from Mumfords book about GIT,...

2
votes

1
answer

94
views

### Is the orbit foliation of the Weyl chamber flow Riemannian?

$\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(...

1
vote

0
answers

45
views

### Reference for rigidity of higher rank action

I heard some results about the rigidity of higher rank action and it looks very interesting. I would like to know if there are any good survey of paper to get started in this field. Thank you in ...

9
votes

1
answer

356
views

### Quotients of schemes by connected groups

Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...

1
vote

0
answers

60
views

### A weakening of infinite Golomb rulers for group actions

If a group $G$ acts on a space $X$, then a Golomb ruler is a subset $A$ of $X$ such that $|gA\cap A|\le 1$ for all $g\in G\backslash\{e\}$.
I am interested in a weaker concept, let's call it a "...

2
votes

1
answer

225
views

### Paradoxical decomposition modulo finite sets

Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there ...

5
votes

1
answer

237
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### Does a compact Lie group action on a family of compact manifolds have diffeomorphic fixed point submanifolds?

Let $\pi: M\to B$ be a fiber bundle of smooth manifolds with $B$ connected and each fiber of $\pi$ is a compact manifold. Let $G$ be a compact Lie group acting smoothly on $M$ such that
$\pi(g\cdot m)=...

1
vote

1
answer

117
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### Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?

When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...

24
votes

1
answer

661
views

### Given a group action on a simplex, can I always find a fundamental region that is a simplex?

Let $\Delta\subset\Bbb R^n$ be a simplex with $n+1$ vertices. Let $G\subset\mathrm{GL}(\Bbb R^n)$ be a finite group of linear symmetries of $\Delta$, i.e. linear transformations that fix the simplex ...

2
votes

0
answers

79
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### Question about finite dimensional representations of a semi-simple Lie group

I have posted a question in MSE
https://math.stackexchange.com/questions/4468138/question-about-finite-dimensional-representations-of-a-semi-simple-lie-group but didn't receive any comment or answer.
...

4
votes

2
answers

218
views

### Special cell decomposition for spheres with free $\mathbb{Z}/p\mathbb{Z}$-action by orthogonal transformations?

Consider the unit sphere $S^d$ in $\mathbb{R}^{d+1}$ with the antipodal action $\nu \colon x\mapsto -x$. This turns $S^d$ into a free $\mathbb{Z}/2\mathbb{Z}$-space.
Construct a CW-complex structure ...

3
votes

0
answers

64
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### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...

7
votes

1
answer

295
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### On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
Is it true ...

1
vote

0
answers

27
views

### Group action in the vicinity of an orbit where the stabilizer jumps

Consider a manifold $M$ with the action of a Lie algebra $\mathfrak g$. Suppose that the action is free,
except for one orbit $O\subset M$ where the stabilizer is a nonzero Lie subalgebra ${\mathfrak ...

2
votes

1
answer

166
views

### Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring

Let $K$ be an algebraically closed field of characteristic zero, and $X$ be an affine $K$-variety (identify $X$ with its set of $K$-points). Let $G$ be group acting "abstractly" on $X$, by ...

2
votes

1
answer

267
views

### Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely

Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...

3
votes

1
answer

185
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### Group action on fibre functor

(I asked this question on mathstack here: https://math.stackexchange.com/questions/4413271/group-action-on-fibre-functor. After getting no response and being suggested in the comments to post it here, ...