# Questions tagged [group-actions]

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327
questions

**4**

votes

**1**answer

127 views

### Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an ...

**2**

votes

**0**answers

113 views

### Why is faithful actions called faithful and who first called it faithful?

Sorry for this question. I asked this on MSE and hsm but no one answered and I decided to post it here that is full of experts.
I want to know why is faithful actions called faithful and who first ...

**3**

votes

**1**answer

115 views

### action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$.
Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via
$$\pi(e_i \wedge e_j) ...

**2**

votes

**0**answers

99 views

### Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\cdot \big(v,\,w\big)\...

**4**

votes

**2**answers

133 views

### Extend (Lie) group action from the boundary to the entire manifold

Let $M$ and $W$ be smooth manifolds such that $\partial W=M$. Let $G$ be a group acting on $M$.
Can one generally extend the action of $G$ to $W$? If not, under which conditions on $W$ and/or $G$ ...

**4**

votes

**2**answers

284 views

### Riemannian manifolds which admit a smooth free $\mathbb{Z}/3\mathbb{Z}$ action but do not admit an equilateral triangle action

A free action of $\mathbb{Z}/3\mathbb{Z}$ on a Riemannian manifold $(M, g)$ is called an equilateral action if for every $x\in M$ all three points of orbit of $x$ have the same distance from each ...

**3**

votes

**0**answers

53 views

### Infinitesimal orbit type decomposition of Hamiltonian $G$-manifolds

Let $G$ be a compact connected Lie group acting in a Hamiltonian fashion on a symplectic manifold $M$ with momentum map $\mu:M\to \mathfrak{g}^\ast$, where $\mathfrak{g}$ is the Lie algebra of $G$. ...

**1**

vote

**0**answers

46 views

### Orbit calculation for normaliser when orbits under centraliser action is known

I was solving a problem and in the middle, I came across this. I will be really grateful for any help here. In the following, we fix the integer $s\ge 2$.
Let $U\le \operatorname{GL}(2^s,\mathbb Z)$...

**2**

votes

**0**answers

53 views

### Articles about Mather's Geometric Groups?

I'm trying to find some information about "Mather's Geometric Groups". But the information on that subject is quite scarce, the only thing I found was the "Mather's Geometric Lemma" in the book "Local ...

**3**

votes

**0**answers

55 views

### When is the symplectic reduction of an action reduced?

Let $X$ be a smooth affine variety with an action of a reductive linear algebraic group $G$ over the complex numbers. We have a moment map $\mu:T^*(X)\to \mathfrak{g}^*$ given by $\mu(x,\xi,Y)=\xi(\...

**5**

votes

**1**answer

242 views

### Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is ...

**2**

votes

**1**answer

149 views

### Equivariant Coefficient ring action on singular cohomology

Let $X$ be a manifold acted on by a Lie group $G$. The $G$-equivariant cohomology of $X$ with coefficients in a ring $\mathcal{R}$ is defined as the cohomology ring
$$
H_G^*(X; \mathcal{R}) := H^*(X_G;...

**6**

votes

**1**answer

195 views

### Free linear group actions on spheres with “strong” angle preservation

Suppose $G$ is a finite group and that $\rho: G\rightarrow O(d)$ is a faithful orthogonal representation, with action on $\mathbb{R}^d$ denoted $\cdot$. Let's say that $\rho$ is "strongly" angle ...

**2**

votes

**0**answers

54 views

### Cohomological dimension of closed $G$-invariant subspaces on homology manifolds with a group action

Suppose $G$ is a compact topological group acting on an $m$-homology manifold $M$ over some ring $R$ by homeomorphisms.
Assume that the action of $G$ is effectively finite on a closed $...

**7**

votes

**2**answers

249 views

### What does the free action of a surface group on an R-tree look like?

Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, ...

**0**

votes

**0**answers

177 views

### Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$

Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...

**4**

votes

**0**answers

80 views

### Amalgamated subgroup of an HNN extension finitely generated

Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated.
Similarly, by ...

**1**

vote

**0**answers

55 views

### Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind

Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...

**2**

votes

**0**answers

156 views

### Finite index subgroup of HNN extension

Let $GX$ be a tree group (a right-angled Artin group such that the graph is a tree), such that not all the subgroups of $GX$ are necessarily RAAGs, so the length of the tree is greater or equal than $...

**0**

votes

**0**answers

52 views

### Finite cyclic group action on Kähler manifold

Let us have a Kähler manifold $M,$ and a smooth action $\phi$ of $\mathbb{Z}/k$ on it, preserving the Kähler structure of it. Then the fixed set of this action $$\text{Fix}(\phi)=\sqcup_{\alpha \in A} ...

**13**

votes

**2**answers

595 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 ...

**4**

votes

**1**answer

451 views

### Is a free and discrete group action on the plane a covering space action?

Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that
every orbit of $G$ is a discrete subset in $\mathbb{R}^2$
$G$ acts freely: ...

**-3**

votes

**1**answer

86 views

### Orbit size of an element [closed]

Let $H$ be a normal subgroup of $G$ and assume that $G$ is acting over a set $X$. Let $c$ be some element of $X$, is there any relationship among the size of the orbit of $c$ under the action of $H$ ...

**10**

votes

**2**answers

652 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 ...

**2**

votes

**1**answer

96 views

### Actions that become free after quotienting out their kernel

Let $H$ be the kernel of an action of a group $G$ on a space $X$. Is there a term for the actions with the property that the action of the quotient group $G/H$ on $X$ is free?

**4**

votes

**0**answers

91 views

### Finite transitive linear subgroups

Let $q$ be a prime power and $d$ an integer. I want to understand the classification of the transitive linear subgroups of $GL_d(\mathbb F_q)$. According to the Wikipedia page https://en.wikipedia.org/...

**4**

votes

**1**answer

205 views

### Homotopy type of G-CW-structure

Let $G$ be a finite group, and $X$ be a free $G$-space. Moreover, assume that $X$ has a homotopy type of a CW-complex. Does $X$ have $G$-homotopy type of a $G$-CW complex also?
Edit: My main ...

**0**

votes

**0**answers

172 views

### Quotient of an affine scheme by an étale finite group

Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)...

**1**

vote

**0**answers

40 views

### Positive Ricci curvature on biquotients

I am working with biquotients and positive curvatures and I was able to give a relatively simple proof for the following:
Theorem: Let $G$ be a compact connected Lie group with a bi-invariant metric $...

**8**

votes

**1**answer

287 views

### Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...

**2**

votes

**1**answer

82 views

### The space of complex structure compatible with metric

Why the space of all complex structure on a $2n$-dimensional vector space which compatible with a positive definite metric is diffeomorphic to $ O(2n)/U(n) $ ?

**0**

votes

**0**answers

231 views

### A question concerning some group action

Let $G$ be a finite group. Consider the set
$$X = \bigcup_{H \le G} G/H$$
which is a disjoint union of left cosets of subgroups $H$ of $G$.
Then $G$ acts on $X$ by left multiplication, and the number $...

**4**

votes

**1**answer

220 views

### Connected permutation groups and wreath product

Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-...

**5**

votes

**3**answers

420 views

### Fixed points under a finite group action on projective variety

Let us have an algebraic action by a finite group G on a complex projective variety $X=\bigcup\limits_{i=1}^N X_i$, whose irreducible components $X_i$ are all smooth and of the same dimension $d$, and ...

**2**

votes

**0**answers

76 views

### Extending linear algebraic group action

Suppose we have a linear algebraic group $G$ acting birationally on a complete variety $X$ i.e. we are given a rational morphism $G\times X\dashrightarrow X$ satisfying obvious properties. Is there a (...

**1**

vote

**0**answers

88 views

### Find representation set of orbits when group acts on a set

Let group $G$ acts on a set $S$. Burnside's lemma gives as how to count numbers of orbits. I am interested how to find the orbits. By finding orbits I mean how to find a representative from each orbit....

**1**

vote

**1**answer

224 views

### Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...

**1**

vote

**1**answer

116 views

### Classification of all equivariant structure on the Möbius line bundles

Is there a classification of all equivariant structures of the Möbius line bundle $\ell\to S^1$?.
For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total ...

**1**

vote

**0**answers

104 views

### Equivariant vector bundles whose quotient map preserves the stabilizer

Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question.
Assume that $E\to M$ is a vector bundle which has the potential of admitting ...

**3**

votes

**2**answers

330 views

### Is this a submanifold?

Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by:
$$\psi : G \...

**9**

votes

**1**answer

207 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

377 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 $\mathrm{GL}_n(\mathbb C)$...

**8**

votes

**1**answer

267 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

99 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

205 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 ...

**3**

votes

**3**answers

208 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

193 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

140 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

226 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 ...

**20**

votes

**1**answer

599 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 $...