# Questions tagged [group-actions]

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**9**

votes

**1**answer

162 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 ...

**3**

votes

**2**answers

367 views

### Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$

For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...

**0**

votes

**0**answers

110 views

### What insight does a regular group-action of a linear algebraic group on an algebraic variety give about the isolated points of the latter?

I'm trying to gather clues on solving this bigger problem (MO link).
So, say I have an algebraic variety $\Omega$ over a field $F$ (e.g $\mathbb R$, $\mathbb C$, $\mathbb F_2$, $\mathbb F_q$).
...

**8**

votes

**1**answer

234 views

### Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...

**2**

votes

**0**answers

95 views

### Projective G-group

Let $G$ be a fixed group.
Can there be projective $G$-groups which are not free $G$-groups?
If yes, for which groups $G$ it happens?
By a "projective $G$-group", I mean a projective object in the ...

**5**

votes

**1**answer

175 views

### Nielsen-Schreier with operations

The Nielsen-Schreier theorem states that subgroups of a free subgroup are free.
Is this hold also for groups with operations?
Explicitly, let $G$ be a fixed group. Let $F$ be a group with $G$-action ...

**2**

votes

**2**answers

136 views

### Computational complexity of sizes and number of orbits of a group acting on a set

I'm interested in the relation between the computational complexity of counting orbits and counting elements in orbits for groups acting on sets. More formally:
Assume that $X_n$ is a infinite ...

**2**

votes

**1**answer

127 views

### Strongly Continuous Group Actions on the $ C^{\ast} $-Algebra of Compact Operators on a Hilbert Space

Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous ...

**4**

votes

**1**answer

91 views

### Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...

**5**

votes

**1**answer

172 views

### Non-Hamiltonian actions in physics

I was reading the following article when I came across the interesting sentence
"non-Hamiltonian [symplectic group] actions also occur in physics"
I took a cursory look at the article cited but ...

**19**

votes

**1**answer

542 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 $...

**6**

votes

**0**answers

141 views

### Which polynomials in the minors of a matrix are invariant under conjugation?

$\newcommand{\Cof}{\operatorname{cof}}$
This is a cross-post.
Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...

**0**

votes

**1**answer

77 views

### Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO.
The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...

**3**

votes

**0**answers

55 views

### Circle actions on simply connected spin manifolds

Recently I've been stuck by a concrete problem. I'll try to make it more general.
Suppose $M$ is a simply connected spin manifold (with higher enough dimension), and $S^1$ acts on $M$ effectively. ...

**3**

votes

**1**answer

98 views

### Operation of a p'-group on a set of p-power order and fix points

The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....

**5**

votes

**1**answer

78 views

### (Euclidean) open orbit in an irreducible real algebraic set

Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real ...

**2**

votes

**1**answer

207 views

### Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$

I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...

**8**

votes

**2**answers

309 views

### Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?
Consider the ...

**1**

vote

**0**answers

65 views

### Volume form preserved by the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $?

I know this is quite an elementary question but I am not an expert in Lie theory.
Does the action of $\mathrm{PGL(n+1, \mathbb{R}})$ on $\mathbf{P}^n(\mathbb{C}) \setminus \mathbf{P}^n(\mathbb{R}) $ ...

**1**

vote

**0**answers

55 views

### sequence definition of proper group action

My understanding is that for an action by a Lie group $G$ on a second countable and Hausdorff differentiable manifold $M$ to be proper, it suffices to show that the map $G \times M \rightarrow M \...

**5**

votes

**1**answer

191 views

### Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?

A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$.
An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...

**8**

votes

**1**answer

208 views

### $\mathbb{C}^{*}$-actions on Fano $3$-folds

I am looking for an example of a smooth Fano $3$-fold $X$ over $\mathbb{C}$, with a non-trival $\mathbb{C}^{*}$-action, which satisfies the following properties:
There is a $\mathbb{C}^{*}$-action ...

**2**

votes

**0**answers

61 views

### Effective actions by non-commutative groups have non-commuting fundamental vector fields?

I have a bit of a contradiction in my brain and I was hoping once again that excellent Mathoverflow community could help me out :)
Let $\rho_g$ be the action associated to a non-abelian Lie Group $G$ ...

**0**

votes

**1**answer

103 views

### Ordered group acting freely on partially ordered set

Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s_1 &...

**1**

vote

**0**answers

108 views

### A question concerning a short exact sequence with an action

Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$.
Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a ...

**2**

votes

**0**answers

61 views

### Does this condition imply symplectic birational cobordism?

From the definition of symplectic birationality given here (https://arxiv.org/pdf/0906.3265.pdf, Definition 2.1), two compact symplectic $2n$-manifolds $(M_{1},\omega_{1}),(M_{2},\omega_{2})$ are ...

**3**

votes

**1**answer

268 views

### Unclear construction in a paper of Ornstein and Weiss

I originally posted this on math.stack, but no one answered, so im posting here:
I need help understanding the following construction (Taken from the paper "Entropy and isomorphism theorems for ...

**14**

votes

**3**answers

372 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 ...

**5**

votes

**0**answers

152 views

### closed substack of quotient stack

The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...

**1**

vote

**0**answers

90 views

### Torus action on hypersurface of multidegree $(1,1,1,1)$ in $(\mathbb{CP}^{1})^{4}$

Let $X$ be a (smooth) hypersurface of mulidegree $(1,1,1,1)$ in $(\mathbb{CP}^{1})^{4}$. According to table $6$ (page 19) in the following paper https://arxiv.org/pdf/1508.01089.pdf, we have:
$X$ has ...

**10**

votes

**1**answer

388 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 ...

**6**

votes

**1**answer

407 views

### Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...

**1**

vote

**1**answer

146 views

### Orbits of unipotent groups over local fields are closed?

Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is ...

**4**

votes

**1**answer

71 views

### Sufficient conditions for secondary invariants

Let $G$ be a finite group, $k$ be a field whose characteristic divides $|G|$, and $\rho:G\hookrightarrow\operatorname{GL}_n(k)$ be a faithful representation of $G$. Let $V$ be a $k$-space of dimension ...

**5**

votes

**0**answers

115 views

### Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...

**3**

votes

**1**answer

143 views

### Is there an easy example of group action where the slice theorem produces a non-trivial principal bundle?

Let $\rho$ be a group action by a compact group $G$
\begin{equation}
\rho:G\times M \rightarrow M \\
\rho:(g,p) \rightarrow \rho_g(p)
\end{equation}
Denote the orbit of $p\in M$ by $\...

**13**

votes

**2**answers

430 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 $V$ ...

**5**

votes

**0**answers

174 views

### Recovering SU(2)-space when the orbit space is a 3 sphere with 3 singular orbits

Background: Consider $SU(2)$ action on the 6-dim flag manifold $M=SU(3)/T^2$ via left multiplication. We view $SU(2)$ as a subgroup of $SU(3)$ corresponding to $2\times 2$-block. The action is just ...

**0**

votes

**0**answers

110 views

### Quotient by augmentation ideal

Let $p$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and let $R = \mathbb{Z}_p [[X_1, \ldots, X_n]] / (f_1, \ldots, f_d)$.
Assume that a finite abelian group $G$ of order ...

**6**

votes

**0**answers

122 views

### Existence of $G$-map between finite $G$-simplicial complex

Let $X, Y$ be finite free $G$- simplicial complex. What kind of properties are necessary for existence a $G$-map,i.e, a continuous map which preserves $G$-action, from $X$ to $Y$? Does existence of ...

**1**

vote

**1**answer

104 views

### Dualizing the trivial action on a $C^*$-algebra

Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T_g(a)=a$ for all $g\in G$ the trivial ...

**2**

votes

**0**answers

34 views

### On the minimum distance along an orbit

Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...

**1**

vote

**0**answers

151 views

### free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$

I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...

**6**

votes

**1**answer

338 views

### Action of upper triangular matrices

Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$.
Do there exist two upper triangular matrices ...

**2**

votes

**1**answer

105 views

### Hamiltonian Group action with infinitely many stabiliser types

What is an example of a connected symplectic manifold $(M,\omega)$, with a Hamiltonian action of $G = U(1) =S^{1}$ with infinitely many stabiliser types?
Infinitely many stabiliser types means that ...

**3**

votes

**0**answers

178 views

### Group acting freely on tree

A tree is a connected acyclic (symmetric) graph. A group acts freely on a graph if there are no inversion of edges and stabilizers of vertices are trivial.
The Bass-Serre Theorem states that A group ...

**2**

votes

**1**answer

240 views

### Action on a normal subgroup where each coset acts freely

Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case ...

**10**

votes

**1**answer

496 views

### 6-manifolds admitting SO(3) action with 2 orbit types

Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose ...

**1**

vote

**0**answers

154 views

### Does a quotient group $G/N$ have a natural action on the regular representation of $G$?

Let $G$ be a group. I am happy to assume niceties such as finite and abelian, but perhaps it is not necessary to answer my question.
Consider the $|G|$-dimensional vector space $V$ (over some nice ...

**3**

votes

**2**answers

174 views

### Affine connections as equivariant maps

An affine torsion-free connection on a smooth manifold $M$ may be thought of as a section of an affine bundle whose associated vector bundle is $S^2(T^*M)\otimes TM$. One would think that this affine ...