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### Transitive action on the sphere

Hello!
From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...

**1**

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**1**answer

257 views

### Orbits of Thompson's group

Thompson's group may act by homeomorphisms on the circle.
Has this action a fixed point?

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**0**answers

91 views

### inertia stratification

Let $X$ be a nice algebraic variety (say smooth, projective) over a field of characteristic 0. Let $G$ be an abelian group acting on $X$. For each subgroup $H$ of $G$, denote by $X^H$ the closed ...

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83 views

### Smoothing of a hyperquotient singularity

Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic ...

**0**

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**1**answer

114 views

### The description of Hurwitz groups

Let $G$ be a Hurwitz group, i.e the automorphism group of some Hurwitz surface $C$. Then Hurwitz's automorphisms theorem shows that the quotient map of $C$ by $G$ has ramification points of indexes ...

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**1**answer

172 views

### On the group actions on Hurwitz surfaces

Let $C$ be a Hurwitz surface, $G=\text{Aut}(C)$ and $N$ is a proper normal subgroup of $G$. Is there a simple argument (without using of classification theorems) for the fact that $N$ acts on $C$ ...

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**1**answer

151 views

### Representations of Finite Subgroups on Homology

Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...

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**3**answers

785 views

### Relation between groups and classifying spaces

Let $G$ be a nonabelian group, with classifying space $BG$.
Motivation: We can compute its homology, $H_\ast(BG)=H_\ast(G)$. It would be nice to see some equivariant computations, like $H_\ast^G(BG)$ ...

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**1**answer

183 views

### Is any $G$-set a coset geometry (in the sense of Tits-Buekenhout)?

Hi there!
Let $X$ be a left $G$-set, and $\Delta=${$x_1,\ldots,x_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx_1,\ldots,Gx_n$} is the orbit space ...

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343 views

### Fixed point of $S^1$-action using roots of unity

Fact: For any (continuous) $S^1$-action on the closed unit disk $\mathbb{D}^n$, there is a fixed point $x_0\in\mathbb{D}^n$.
I have thought of a possible argument that re-proves this, but am not sure ...

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**2**answers

202 views

### On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...

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216 views

### 2 questions on Nagata's counterexample; $k[f_1,…,f_r]=k[g_1,…,g_s]$ vs. $k(f_1,…,f_r)=k(g_1,…,g_s)$

Let $\{a_{ij}\}$ for $i=1,2,3$, and $j=1,...,16$ be algebraically independent elements over some prime field. Let $k$ be a field containing all $a_{ij}$. Then consider $k^{16}$ as $k$-vector space and ...

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**1**answer

206 views

### Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.
Recall that an ...

**1**

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**1**answer

351 views

### Free and cellular G-action implies free G-complex?

Recall that a CW-complex $X$ with an action of a group $G$ which permutes the cells (i.e., for any $g \in G$ and any cell $\sigma \subseteq X$, $g\sigma$ is a cell) is called a $G$-complex. If the ...

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**1**answer

347 views

### Free group actions on varieties and algebras of coinvariants

Suppose $k$ is an algebraically closed field of characteristic zero and $A$ is a finitely generated commutative associative reduced $k$-algebra.
Suppose the group $\mathbb{Z}_2$ acts on $A$ in such a ...

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396 views

### “Homogeneity” of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]

My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...

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**1**answer

198 views

### volume of exceptional group orbits

Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by ...

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**5**answers

622 views

### Examples of manifolds with effective circle actions?

I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d ...

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**2**answers

770 views

### Symmetric group action on squarefree polynomials

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest.
Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...

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**1**answer

137 views

### Find an action of $\mathbb{Z}/2$ on $\mathbb{C}P^1$ which is compatible with the fraction linear transform of $SL(2,\mathbb{R})$

There is a natural fraction linear transform of $SL(2,\mathbb{R})$ on $\mathbb{C}P^1$ given by:
$$
\begin{pmatrix} a & b \\
c & d \end{pmatrix} \cdot[z,w]=[az+bw,cz+dw].
$$
Let ...

**12**

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**3**answers

724 views

### Actions on Sⁿ with quotient Sⁿ

What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?
Comments.
I am mostly interested in (maybe trivial) properties of such ...

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**0**answers

424 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

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**1**answer

338 views

### Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open?

It is well-known that if a reduced algebraic group $G$ acts on a separated reduced scheme $X$, and $G$ acts trivially on a dense open subscheme $U\subseteq X$, then the action is trivial.
If $X$ is ...

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313 views

### Invariant for group actions

Hello everybody!
Define the action of $SL_4({\mathbb{Z}})$ on alternating 2-forms or simply skew-symmetric matrices of degree 4 according to the following:
For $B \in SL_4({\mathbb{Z}})$ and an ...

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**2**answers

279 views

### Group action on spin^c 4-manifold.

[edit]
I'll try to be more precise.
In paper N.Nakamura, "Bauer–Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means:
...

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**1**answer

218 views

### eigen-bundles of a trivial vector bundle

Suppose I have a trivial vector bundle $V\cong \mathcal{O}_C^{\oplus s} \rightarrow C$ on an algebraic variety $C$, and suppose furthermore that I have an action $\mu$ of a cyclic finte group $G$ on ...

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291 views

### Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group?

The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal ...

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**1**answer

524 views

### Abstract definition of properly discontinuous action

A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract ...

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**0**answers

89 views

### invariant lines avoiding fixed subvarieties

Could anybody help me with the following question ?
Assume we are given:
(1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$,
(2) a closed algebraic ...

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**1**answer

346 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 $n$?

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**1**answer

350 views

### Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...

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124 views

### Minimality of time-t minimal flows

This question is mainly motivated by the question Transitivity of a flow and its time-1 map
Let $M$ be a closed smooth manifold and $\Phi\colon\mathbb{R}\times M\to M$ be a smooth minimal flow, i.e. ...

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**1**answer

401 views

### Free $\mathbb{Z}_2$-actions match at some point

I have in front of me a proof of this lemma:
If $f$ and $g$ are free $\mathbb{Z}_2$-actions on $S^1$, then $f(x)=g(x)$ for some $x \in S^1$.
A $\mathbb{Z}_2$-action on the unit circle $S^1$ is a ...

**3**

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**1**answer

321 views

### Action on a compact group

If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?

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**1**answer

267 views

### a question on continuity of $G$-module for a profinite group $G$

I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a ...

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**2**answers

294 views

### $S^1$-action in three dimensions

Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action.
What does this imply for $M$? What are examples except for (products of) spheres?

**1**

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**1**answer

255 views

### Does the semi-stable set determine the linearization of a GIT quotient?

Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group.
Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the ...

**1**

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**1**answer

262 views

### is a closed subscheme of the projective line closed under the action of Gal(Qbar/Q)

Let $S$ be a non-empty closed subscheme of $P^1_K$, where $K$ is a number field. Assume that the cardinality of $S$ is finite.
Is $S$ closed under the action of the absolute Galois group of the ...

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votes

**4**answers

978 views

### What about the empty torsor?

Let $G$ be a group. A $G$-torsor is a set $X$ together with an action of $G$ such that for all $x,y \in X$ there is exactly one $g \in G$ such that $gx=y$. This looks like a group which has forgotten ...

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512 views

### Untwisting the Cohomology with Twisted Coefficients

This question is set on a finite $2$-group $G$ and a subgroup $H$ of index $2$ (but perhaps the question could be answered for arbitrary orders/indexes).
It was asked here on MO whether ...

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**1**answer

391 views

### A question regarding Lie group actions

Can you give me an example of a Lie group acting on a compact metric connected space transitively so that it has a closed finite index subgroup which does not act transitively?

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5k 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 ...

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**0**answers

180 views

### Faithful actions of finite groups on topological spaces

Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...

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1k views

### Determinant associated with a group action

Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries ...

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**1**answer

198 views

### Does an abelian group acting on a riemaniann manifold define an othogonal foliation?

This question is related to my previous question. Suppose that a group $G$ acts freely and properly on a Riemaniann manifold $(M, g)$. Than the orbits form a foliation for $M$. For $p \in M$, let ...

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**1**answer

456 views

### Actions of $Z_n$ and actions of $Z_{n-1}$

I am playing with some questions concerning connections between
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.
When ...

**10**

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**4**answers

932 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 ...

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4k 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 ...

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**10**answers

2k views

### Looking for interesting actions that are not representations

As a person interested in group theory and all things related, I'd like to deepen my knowledge of group actions.
The typical (and indeed the most prominent) example of an action is that of a ...

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**2**answers

185 views

### The antipodal action on a connected one dimensional manifold

When I am reading one paper, I have met the following statement:
It is impossible to define a $Z_{2}\times Z_{2}$ action on a connected closed curve on a compact Riemann surface.
The claim is ...