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**22**

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 ...

**0**

votes

**0**answers

54 views

### Action of semidirect products of cyclic groups

Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...

**5**

votes

**2**answers

231 views

### Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
Suppose there exists some point $x \in X$ whose ...

**1**

vote

**1**answer

157 views

### finite stabilizers + compact orbit space => proper action?

Suppose a countable discrete group is acting on a smooth manifold with finite stabilizers and the orbit space is compact (and Hausdorff). How one can prove that the action is proper?
I found this ...

**1**

vote

**0**answers

56 views

### Name for condition on map of cancellative monoids

Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such ...

**1**

vote

**1**answer

86 views

### The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...

**4**

votes

**1**answer

104 views

### Are epimorphisms (defined via an obvious action) of free Boolean algebras whose set of generators is a group automorphisms?

Let $G$ be a group. Consider $B$, the free Boolean algebra with generating set (I'll call them "variables") $G$. Let $F$ be some formula (that is, some fixed element of $B$). Define an endomorphism ...

**1**

vote

**2**answers

180 views

### Connected components of algebraic groups

Let $G$ be an algebraic group, and $G_{Id}$ the connected component of the identity. Then $G_{Id}$ is a normal subgroup of $G$ and $G/G_{Id}$ is the component group of $G$.
Let $G_{c}\subset G$ be ...

**1**

vote

**1**answer

130 views

### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain ...

**5**

votes

**2**answers

174 views

### fixpoint algebras of a permutation action

Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M_k(\mathbb{C})$. The permutation group $\Sigma_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a ...

**6**

votes

**0**answers

138 views

### Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?

Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...

**6**

votes

**1**answer

694 views

### Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...

**8**

votes

**1**answer

302 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 ...

**5**

votes

**1**answer

337 views

### Faithful transitive actions by large groups on small sets

How large is the largest transitive subgroup of $S_n$ other than itself and $A_n$? In particular, does its size grow at least exponentially in $n$?

**11**

votes

**1**answer

347 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?

**3**

votes

**2**answers

205 views

### 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 ...

**2**

votes

**0**answers

158 views

### Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq ...

**6**

votes

**1**answer

148 views

### Is there a highly transitive action of a finitely generated torsion simple group?

Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?
Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two ...

**0**

votes

**1**answer

184 views

### Coaction of a group

Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I ...

**6**

votes

**1**answer

219 views

### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

**4**

votes

**1**answer

288 views

### Free action of $\mathbb{Z}(2^{\infty})$ on a compact space

Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on ...

**1**

vote

**0**answers

90 views

### Adelic integral factorization

In order to calculate Tamagawa numbers, I need to justify that for a nice (say Schwartz-Bruhat) function, the following identity holds :
$$\int_{\mathbf{A}^2} f(x)dx = \int_{SL_2(\mathbf{A})/SL_2(K)} ...

**5**

votes

**0**answers

190 views

### Unitary representations of Tarski Monsters and other beasts

Did people study the unitary representations of Tarsky Monsters, for example the ones constructed by Ol'shanskii? Are there any exotic representations, ie. except the ones related to the left regular, ...

**0**

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**0**answers

57 views

### Invariant subsets of a local action

I have also asked this in MSE, but it seems to me that my question wasn't very well received there and I think someone in here will be able to answer it more quickly, hence this post.
I don't ...

**1**

vote

**0**answers

167 views

### 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 ...

**8**

votes

**1**answer

303 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 ...

**33**

votes

**9**answers

3k 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 ...

**-2**

votes

**0**answers

175 views

### Relations between Arboreal Group Theory and Tree Group Actions? [closed]

By a tree group action, we mean an action of a group $G$ over the infinite regular binary tree $T_2$ such that for each $g \in G$, the mapping $x \mapsto g\cdot x$ is an automorphism of $T_2$; these ...

**4**

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**0**answers

220 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 ...

**1**

vote

**1**answer

157 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 ...

**2**

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**0**answers

155 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 $f:S^{n}\to \mathbb{R}^{n}$ ...

**2**

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**0**answers

105 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 ...

**1**

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**0**answers

113 views

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

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...

**13**

votes

**4**answers

1k 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 ...

**2**

votes

**1**answer

157 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 ...

**4**

votes

**0**answers

121 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 ...

**2**

votes

**1**answer

100 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 ...

**-1**

votes

**1**answer

92 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 ...

**0**

votes

**0**answers

73 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 ...

**4**

votes

**2**answers

269 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 ...

**7**

votes

**5**answers

4k 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 ...

**16**

votes

**2**answers

528 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 ...

**3**

votes

**0**answers

113 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 ...

**7**

votes

**1**answer

249 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 ...

**4**

votes

**0**answers

229 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)\::\: ...

**1**

vote

**1**answer

74 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 ...

**6**

votes

**1**answer

174 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 ...

**10**

votes

**4**answers

906 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 ...

**6**

votes

**1**answer

219 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 ...

**5**

votes

**1**answer

425 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} ...