Questions tagged [group-actions]

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4
votes
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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 $...

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