Questions tagged [group-actions]

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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(...
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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 ...
  • 195
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 ...
  • 575
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)=...
  • 8,193
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 ...
  • 10.6k
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. ...
  • 53
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 ...
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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 ...
  • 3,399
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 ...
  • 19.3k
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 ...
  • 111
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 ...
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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 $\mathbb{R}^3$?
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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 ...
  • 19.6k
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 ...
  • 53
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 ...
  • 53
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 $...
  • 195
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 ...
  • 161
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 $|G|$...
  • 433
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:...
  • 405
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 ...
  • 53
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, $$ (...
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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 ...
  • 710
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 ...
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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 $...
  • 1,439
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 ...
  • 6,508
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{...
  • 4,819
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 ...
  • 53
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 + &...
  • 53
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 ...
  • 59
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 $C_n(\mathbb{R}^d)$ ...
  • 3,773
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 ...
  • 6,508
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 ...
  • 142
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 $\...
  • 1,639
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 ...
  • 405
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 ...
  • 2,706
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 ...
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