Questions tagged [group-actions]

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67 votes
3 answers
20k views

Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
Martin David's user avatar
  • 1,216
65 votes
9 answers
9k views

List of Classifying Spaces and Covers

I am looking for a list of classifying spaces $BG$ of groups $G$ (discrete and/or topological) along with associated covers $EG$; there does not seem to be such cataloging on the web. Or if not a ...
31 votes
7 answers
5k views

Invariant polynomials under a group action (hidden GIT)

Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$). Now the symmetric group $\mathfrak{S}_n$ ...
babubba's user avatar
  • 1,953
29 votes
4 answers
7k views

Intuitive explanation of Burnside's Lemma

Burnside's Lemma states that, given a set $X$ acted on by a group $G$, $$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$ where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of ...
Zev Chonoles's user avatar
  • 6,702
24 votes
1 answer
684 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 ...
M. Winter's user avatar
  • 11.9k
23 votes
4 answers
1k views

Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$? Here "naturally" means "in an $GL(V) \times GL(W)$-...
James Propp's user avatar
  • 19.1k
23 votes
1 answer
995 views

Finite-order self-homeomorphisms of $\mathbf{R}^n$

Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be of finite order if $\phi^m = \mathrm{id}_{\mathbf{R}^n}$ for some ...
Wille Liu's user avatar
  • 1,036
22 votes
5 answers
27k views

What is the standard notation for group action

Please let me know what is the standard notation for group action. I saw the following three notations for group action. (All the images obtained as G\acts X for ...
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 $|G|$...
Craig's user avatar
  • 525
21 votes
1 answer
671 views

Diameter of a quotient of the infinite dimensional sphere

Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well. Assume that the action $...
Anton Petrunin's user avatar
21 votes
1 answer
548 views

Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series $\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely. Then there is a partition of the symmetric group ${\rm Sym}(\...
Stefan Kohl's user avatar
  • 19.3k
20 votes
5 answers
3k views

How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...
Orbicular's user avatar
  • 201
19 votes
1 answer
1k views

A result on Lie group actions on 15-dimensional spheres?

In this interview by Eric Weinstein to Roger Penrose, Timestamp 1:24:05., what result is the host talking about? Transcription of the relevant part: "If you have two sets of symmetries, known as ...
Qfwfq's user avatar
  • 22.4k
17 votes
1 answer
365 views

Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?

Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma, $$ |V/G| = \frac 1{|G|} \sum_{g\in G} q^{\dim(ker(g - I))}. $$ Since $g-I$ ...
Amritanshu Prasad's user avatar
17 votes
1 answer
493 views

Is a smooth action of a semi-simple Lie group linearizable near a stationary point?

Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
Dick Palais's user avatar
  • 15.1k
17 votes
0 answers
420 views

On manifolds which do not admit (smooth) actions of finite groups

Question: Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no continuous effective group actions of ...
Abenthy's user avatar
  • 517
17 votes
0 answers
568 views

Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer. I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order. Or equivalently, ...
Anton Petrunin's user avatar
16 votes
3 answers
953 views

Actions on Sⁿ with quotient Sⁿ

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$? Comments. I am mostly interested in (maybe trivial) properties of such actions ...
Anton Petrunin's user avatar
16 votes
2 answers
599 views

why most of the angles are right

The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which ...
Anton Petrunin's user avatar
16 votes
1 answer
914 views

Generalizing the Mazur-Ulam theorem to convex sets with empty interior in Banach spaces

The Mazur-Ulam theorem (1932) states that any isometry of a normed linear space is affine. See Nica (Expo. Math. 30 (2012), 397-398; arXiv:1306.2380) for a very elegant proof. Question: Let $M$ be a ...
Mikhail Ostrovskii's user avatar
15 votes
10 answers
5k views

Looking for interesting actions that are not representations

As a person interested in group theory and all things related, I'd like to deepen my knowledge of group actions. The typical (and indeed the most prominent) example of an action is that of a ...
Marek's user avatar
  • 364
15 votes
0 answers
615 views

"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
Renato G. Bettiol's user avatar
14 votes
5 answers
2k views

A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all, So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$ Is there a general ...
Ngoc Mai Tran's user avatar
14 votes
3 answers
506 views

Proving convergence of sum over $\mathbb{Z}^n$

In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the ...
P. May's user avatar
  • 143
14 votes
4 answers
1k views

actions of the hyperoctahedral group

I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
Martin Rubey's user avatar
  • 5,473
14 votes
1 answer
647 views

If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?

This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$ $\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
Anthony Carapetis's user avatar
14 votes
2 answers
627 views

Action that is Bourbaki proper but not Palais proper

I'm working with different definitions of proper action (Cartan, Bourbaki and Palais) and the relation between them. All the spaces I'm working with are $T_{3.5}$, the definitions are: If $U$ and $V$ ...
Marcos TV's user avatar
  • 193
14 votes
2 answers
1k views

Symmetric group action on squarefree polynomials

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
Vidit Nanda's user avatar
  • 15.3k
13 votes
2 answers
688 views

Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?

More precisely, is there a criterion that decides the above question? I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a ...
Caterina C.'s user avatar
13 votes
3 answers
1k views

The classifying space of a gauge group

Let $G$ be a Lie group and $P \to M$ a principal $G$-bundle over a closed Riemann surface. The gauge group $\mathcal{G}$ is defined by $$\mathcal{G}=\lbrace f : P \to G \mid f(p \cdot g) = g^{-1}f(p)...
H. Shindoh's user avatar
13 votes
2 answers
600 views

Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

I'm coming with a very basic question for which I can't find an answer. Please forgive me if I didn't search efficiently enough. What can the closure of an orbit of an element $X$ of $\mathbb{R}^2$ ...
Selim G's user avatar
  • 2,616
13 votes
1 answer
690 views

Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex?

If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex ...
ychemama's user avatar
  • 1,326
13 votes
1 answer
998 views

Category without identities?

Just as a monoid is a category with a single object, a semigroup may be seen as a non-unital category, still with associative composition. Then an $S$-set for $S$ a semi-group can be seen as a functor ...
Arrow's user avatar
  • 10.3k
13 votes
1 answer
1k views

When taking the fixed points commutes with taking the orbits

Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.) The set $\text{Fix}_H(X)$ of $H$-fixed ...
Tom Leinster's user avatar
12 votes
1 answer
422 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
13829's user avatar
  • 121
12 votes
1 answer
1k views

Cobounded ⇒ cocompact?

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and $$\operatorname{diam} H/\Gamma\le 1.$$ Is it true that $H/\Gamma$ is compact? Stupid example. Assume the action of $\...
Anton Petrunin's user avatar
11 votes
2 answers
884 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 $(m+1)$-...
Anton Petrunin's user avatar
11 votes
3 answers
1k views

Quotient of a smooth curve by a finite group and differentials

Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...
Lisa S.'s user avatar
  • 2,623
11 votes
1 answer
426 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 $\...
Jialong Deng's user avatar
  • 1,699
11 votes
2 answers
2k views

Determinant associated with a group action

Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries ...
Denis Serre's user avatar
  • 50.6k
11 votes
2 answers
1k views

Orbifolds vs. branched covers

Forgive me if this is a basic question. I'm just learning about orbifolds, and covering spaces are my happy place for thinking about group actions. If $M$ is a manifold and $G$ is a group acting ...
Taylor McNeill's user avatar
11 votes
1 answer
604 views

Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex

Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$. (Terminology: By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is ...
Michał Kukieła's user avatar
11 votes
1 answer
524 views

Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
Pablo's user avatar
  • 11.2k
10 votes
3 answers
772 views

Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?

I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!
sara's user avatar
  • 247
10 votes
4 answers
1k views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
Kamran Reihani's user avatar
10 votes
3 answers
900 views

Regular subsets of $\text{PSL}(2, q)$

Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ regular (or sharply transitive or simply transitive or...) if for every two points $\omega_1, \omega_2 \in \Omega$ there is a ...
Sean Eberhard's user avatar
10 votes
1 answer
1k views

Białynicki-Birula theory for non-complete varieties

I would like to know to which extent the theory developed for smooth projective varieties in the following articles A. Białynicki-Birula, Some theorems on actions of algebraic groups. Ann. of ...
Qfwfq's user avatar
  • 22.4k
10 votes
2 answers
603 views

What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
Daryl Cooper's user avatar
10 votes
1 answer
538 views

An equivariant social choice in Mathematical economics

Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to ...
Ali Taghavi's user avatar
10 votes
1 answer
230 views

A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$

Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices (A138464 on the OEIS). The first few values of $F(n;i) \pmod n$ are listed below: $$\begin{array}{r|rrrrrrrrrrr} & i=0 &...
Rebecca J. Stones's user avatar

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