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### Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups ...

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442 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 $\mathbb{R}^2$ ...

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**1**answer

363 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 ...

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268 views

### Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...

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**1**answer

165 views

### Free action of symmetric groups

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$?
Is there a compact manifold which can be act freely by all symmetric ...

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161 views

### A possible generalization of the Borsuk Ulam theorem via action of symmetric groups

The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...

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366 views

### Center of one-point stabilizer in 2-transitive groups

In this MO question it was mentioned that the following fact seems to be true:
If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a
non-trivial center, then $G$ is of ...

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111 views

### A topological space extracting from a group action

Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to ...

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382 views

### Non invertibility of certain integral arising from group action

Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question.
Edit 2: I understand from the following post that some part of the previos version of my question has ...

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**1**answer

171 views

### When is Aut(G) the symmetric group of an Aut(G)-invariant generating set?

Let $G$ be a group, $X$ a generating set of $G$. Suppose $X$ is $\operatorname{Aut}(G)$-invariant, i.e. $\sigma(X)\subseteq X$ for all $\sigma \in \operatorname{Aut}(G)$. When is the restriction ...

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129 views

### Fixed sets of orbit spaces

I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?)
Let $G$ be a compact Lie ...

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97 views

### Action of rotation group on Matrices [closed]

Is the following assertion true?
Suppose $p, q \geq 3$. Consider the action of $SO(p,\mathbb{R})$ on $p \times q$ matrices by left multiplication. I want to show that $MA = A$, where $M \in ...

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75 views

### Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...

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119 views

### Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...

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237 views

### Polynomial dynamical systems

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...

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185 views

### Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write
Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups ...

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293 views

### Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...

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122 views

### Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...

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446 views

### A generalized Burnside's lemma

Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:
$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} ...

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202 views

### Is translation by the free group (in two generators) on a certain completion of the group an amenable action?

Let $\mathbb{F}_2 = \langle a,b\rangle$ be the free group in two generators $a,b$ and let $\alpha \in \text{End}(\mathbb{F}_2)$ be given by $\alpha(a) = a^2, \alpha(b)= b^2$. Note that the index ...

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525 views

### Action of a Lie group with finitely many orbits

EDIT: Let a real Lie group $G$ act on a smooth manifold $M$ with finitely many orbits such that each orbit is locally closed ($M$, but not $G$, may be assumed to be compact in my case). Let ...

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290 views

### Recommended textbooks for Hamiltonian group actions?

I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any ...

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**1**answer

281 views

### How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map
$$
\mu: G\times ...

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107 views

### Group action of $G<\mathbb Z^\infty_2$ over the Golden mean shif [closed]

I'm am looking for an action of an infinite subgroup of $\mathbb Z^\infty_2$ over the golden mean shift space $$X=\{x\in \{0,1\}^\mathbb N : x_i=1\Rightarrow x_{i+1}=0\}$$ such that any element of $G$ ...

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510 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 ...

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**1**answer

115 views

### From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...

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128 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 ...

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143 views

### is the fixed point locus integral and reduced?

Let $X$ be a scheme over a field $k$ of characteristic zero and let $G$ be finite group acting on $X$. Then one can define the scheme $X^G$ of fixed points of $X$. It is a closed smooth subscheme of ...

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90 views

### is the exceptional divisor of an equivariant blow-up linearized?

I hope someone can help me with this.
Let $X$ be a smooth projective variety, say over the complex numbers and let $Z$ be a closed subvariety of $X$. Assume that $X$ is acted upon by a finite group ...

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### A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$

Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of ...

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109 views

### Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space

Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...

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81 views

### 'Convex' slices of proper actions

Consider a Lie group $G$ acting properly on a manifold $M$. Then by the slice theorem we can find for any point $m\in M$ a submanifold transverse to the orbit $\mathcal{O}$ through $m$ and which is ...

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293 views

### Action of Mapping Class Group on Arc complex

Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...

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601 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 ...

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607 views

### Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...

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145 views

### Diffeomorphism between open annuli preserving common symmetries

Suppose $A$ and $B$ are subsets of $\mathbb{R}^2$ homeomorphic (and thus $C^\infty$ diffeomorphic) to the open annulus (punctured $\mathbb{B}^2$) and let $G$ be the group of isometries of ${\mathbb ...

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163 views

### Is the limit set of a group action always closed?

Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in ...

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421 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 ...

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84 views

### Ergodic actions with co-finite stabilizers

Let $G$ be a locally compact, second countable group acting on a standard probability space $(X,\nu)$, and let $\nu$ be $G$-invariant. Let $G_x = \{g \in G\,:\, gx=x\}$ denote the stabilizer of $x \in ...

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538 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 ...

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**1**answer

140 views

### Tensor Powers of 1-Dimensional Representations of a Finite Group

Let $G$ be a finite group acting on a commutative ring $R$ via ring maps. In doubt, one can assume $R$ to be noetherian or regular if one wants. Let $P$ be a $1$-dimensional free $R$-module with a ...

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167 views

### Counterexamples to analogue of Cannon conjecture in higher dimensions

It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for ...

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291 views

### Fixed points on boundary of hyperbolic group

Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?

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101 views

### Actions of compact Lie groups on (possibly but hopefully not) non-regular spaces

Suppose $G$ is a compact Lie group acting freely on a topological space $Q$ (about whose separation conditions I make no assumptions) and the qoutient $Q/G$ is known to be completely regular Hausdorff ...

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255 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 $g-I$ ...

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142 views

### When a Whitney stratification has no stratum of codimension one?

Let $G$ be a compact Lie group, and $M$ be a smooth $n$-dimensional $G$-manifold which admits an orientation preserving the $G$-action. Then $M$ has a natural Whitney stratification induced by the ...

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266 views

### Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...

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125 views

### automorphisms acting trivially on projective spaces

Let $K$ be a field and $\mathbb{P}=\mathbb{P}^n_K$ the projective space of dimension $n$ over $K$. Consider a linear automorphism $g$ of $\mathbb{P}$.
Is it true that $g^\ast$ acts trivially on ...

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198 views

### blowups and group actions

Let $X$ be a smooth projective variety over the complex numbers and assume that $X$ is equipped with the action of a finite group $G$.
Denote by $Z$ the closed subscheme of fixed points of $G$ and ...

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94 views

### “Spectral decomposition” action on the unitary group

Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$.
What is ...