# Questions tagged [group-actions]

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435
questions

2
votes

1
answer

56
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(...

0
votes

0
answers

36
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

310
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

43
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

215
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

216
views

### 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

111
views

### 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

569
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

70
views

### 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

166
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

57
views

### 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 ...

1
vote

0
answers

45
views

### Example on pseudo-groups

Definition:
A pseudo-group is a collection $\mathcal{G}$ of (locally defined) invertible smooth diffeomorphisms of a manifold $M$. The simplest example of a pseudo-group is the collection of all ...

7
votes

1
answer

251
views

### 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

24
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 ...

1
vote

1
answer

123
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

220
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

143
views

### 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, ...

5
votes

1
answer

85
views

### Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$

Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?

2
votes

1
answer

181
views

### Difference between definitions of continuous action, profinite case

My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could ...

4
votes

0
answers

94
views

### Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...

4
votes

1
answer

186
views

### Integer-valued polynomials from Pólya counting

Let finite group $G$ act on a finite set $X$ and hence on colorings $Y^X$, where $Y=\{1,2,\ldots,k\}$ is a set of colors. The Burnside-Pólya-Redfield-etc. counting theorem says that the number of ...

2
votes

0
answers

117
views

### Fundamental vector field

Definition: Let $M$ be a smooth manifold on which acts a Lie group $G$. Let $X$ be an element in the Lie algebra $\mathfrak{g}$ of $G$, we associate to it the vector field $X_M$ called the fundamental ...

3
votes

1
answer

91
views

### Define a symplectic structure on $G \times_{G_\beta} V$, where $V$ is symplectic

Let $G$ be a compact Lie group with algebra $\mathfrak{g}$. Let $\beta $ be an element in the dual of the Lie algebra $\mathfrak{g}$. We denote by $G_\beta$ the stabilizer subgroup of $\beta$ by ...

3
votes

1
answer

111
views

### Almost free Lie group action

It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G_x$ is finite for any $...

0
votes

0
answers

70
views

### Do there exist methods for determining the orbits of a group action on the cartesian product of sets?

Suppose that we have some group $G$ acting on some set $\Omega$. Then $G$ acts on $\Omega^n = \Omega \times \cdots \times \Omega$ ($n$ times) naturally. I wonder, is there an iterative algorithm to ...

22
votes

2
answers

1k
views

### When does $G\times G\times G$ admit a faithful group action on a set of size $|G|$?

[Edited due to YCor's comment:]
Given a finite group $G$, under what conditions does $G\times G\times G$ (the direct product of three copies of $G$) admit a faithful group action on a set of size $|G|$...

2
votes

1
answer

116
views

### Action of a group $G$ induces a coaction on $C_0(G)$

In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259.
Let $G$ be a locally compact group and $C$ be a $C^*$-algebra. Assume an action
$$\alpha:...

3
votes

1
answer

102
views

### Proof of the Hamiltonian slice theorem

Theorem(Guillemin Sternberg Marle) Let $(M, \omega, \mu) $ be a symplectic manifold together with a Hamiltonian group action. Let $p$ be a point in $M$ such that $O_p $ is contained in the zero level ...

0
votes

0
answers

51
views

### How to show that, $ C_c ( \pi_{1}^{ \mathrm{top} } (X) ) $ is dense in $ C_0 (X/G) $?

Let $G$ be a locally compact, Hausdorff, second countable, topological group.
Let $X$ be a locally compact, Hausdorff $G$-space.
The action of $G$ on $X$ gives an action of $G$ on $ C_0 (X) $ by, $$ (...

1
vote

0
answers

116
views

### Equivariant embeddings

According to the article I got from this question, closed Riemann manifolds together with their isometry groups (which is always a compact Lie group) can be smoothly and equivariently embedded into ...

6
votes

0
answers

134
views

### Fundamental domains for proper Lie group actions on smooth manifolds

The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms.
Motivation: when trying to figure out the homeomorphism type of the orbit ...

4
votes

1
answer

114
views

### The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...

3
votes

1
answer

260
views

### Is an equivariant projective morphism equivariantly-projective?

Let everything be over $\mathbb{C}$. Consider two varieties $X,$ $Y,$ where $X$ is normal and $Y$ is affine,
having regular $\mathbb{C}^*$-actions and
a $\mathbb{C}^*$-equivariant projective morphism
$...

3
votes

0
answers

135
views

### A circle action on the E8 manifold

In the paper "A survey of group actions on 4-manifolds" by
Allan L. Edmonds on page 5 there is the remark "One should note that the coned-off E8-plumbing manifold
admits a circle action ...

6
votes

1
answer

109
views

### Stabilizers of multilinear forms

Let $\{e_1,\ldots, e_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum_{i=1}^n e_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$
and consider the action of $\text{...

1
vote

1
answer

158
views

### Symplectic structure on a vector bundle

Let $G$ be a Lie group and let $M \rightarrow B$ be a $G$-equivariant vector bundle with typical fiber $E$.
Suppose that $B$ and $E$ are both symplectic manifolds with a $G$-Hamiltonian action. Can we ...

4
votes

1
answer

153
views

### Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...

1
vote

0
answers

71
views

### The Pushforward of the Liouville measure

Let's consider a Hamiltonian action of a torus $T$ on a symplectic manifold $(M, \Omega)$. We denote by $\mu: M \rightarrow \mathfrak{t}^*$ the moment map and by $\Omega_\mathfrak{t}(X) := \Omega + &...

6
votes

1
answer

274
views

### Trans-amenability of group actions

This problem is derived from this post.
Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are ...

2
votes

0
answers

127
views

### Fixed point subalgebra

Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...

2
votes

0
answers

31
views

### On the higher-dimensional Berry-Robbins problem

Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...

3
votes

0
answers

77
views

### non-smooth manifold with circle action (with fixed points)

I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...

7
votes

1
answer

242
views

### Is $\mathbb{Q}$ "equivalent" to a structure with transitive automorphism group action?

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For ...

1
vote

1
answer

107
views

### connected components of the fixed point subvariety

Let $X$ be a smooth complex variety with an action of a finite group $G$.
The fixed point subvariety $X^G$ is smooth but may have many connected components.
What determines these connected components ...

10
votes

1
answer

384
views

### Scalar curvature and the degree of symmetry

Let $M$ be a closed connected smooth manifold. We define the degree of symmetry of $M$ by $N(M):=\sup_\limits{g}\mathrm{dim}\,\mathrm{Isom}(M,g)$, where $g$ is a smooth Riemannian metric on $M$ and $\...

3
votes

1
answer

195
views

### Equivalent definitions of strongly proximal action

Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...

2
votes

1
answer

235
views

### Behavior of canonical divisor under a finite group quotient

Given a smooth algebraic surface $X$, and a group $G$ acting on it and letting $Y := X / G$, how can we compute $K_Y^2$ from from $K_X^2$?
Current progress: In Borisov and Fatighenti - New explicit ...

5
votes

2
answers

260
views

### Sections of a polar action are totally geodesic

This question is a repost of the following: https://math.stackexchange.com/questions/4195805/sections-of-a-polar-action-are-totally-geodesic. I've decided it to post it here because it didn't seem to ...

1
vote

0
answers

52
views

### Row-wise conjugation of completely bounded map by group action

Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...

3
votes

0
answers

65
views

### Coordinates for quasiperiodic motion after reconstruction

Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...