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3
votes
5answers
462 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 ...
10
votes
2answers
745 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 ...
1
vote
1answer
134 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 ...
11
votes
3answers
667 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 ...
15
votes
0answers
394 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, ...
3
votes
1answer
297 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 ...
3
votes
2answers
301 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 ...
1
vote
2answers
240 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: ...
0
votes
1answer
189 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 ...
4
votes
2answers
254 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 ...
3
votes
1answer
460 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 ...
1
vote
0answers
87 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 ...
4
votes
1answer
260 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 $n$?
1
vote
1answer
273 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 ...
3
votes
0answers
121 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. ...
7
votes
1answer
382 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
votes
1answer
312 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 ?
1
vote
1answer
227 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 ...
6
votes
2answers
279 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
vote
1answer
211 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
vote
1answer
254 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 ...
4
votes
4answers
937 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 ...
3
votes
0answers
496 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 ...
2
votes
1answer
383 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?
6
votes
4answers
3k 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 ...
0
votes
0answers
165 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 ...
10
votes
2answers
848 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 ...
0
votes
1answer
195 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 ...
1
vote
1answer
448 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 ...
9
votes
4answers
819 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 ...
29
votes
10answers
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 ...
8
votes
10answers
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 ...
1
vote
2answers
179 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 ...
4
votes
2answers
791 views

Terminology: “cocompact”

Let $M$ be a Riemannian manifold such that its isometry group $G=\textrm{Iso}(M)$ is a Lie group, and let $\Gamma$ be a subgroup of $G$. 1) What does the phrase "$\Gamma$ is a cocompact group of ...
25
votes
3answers
6k views

Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
0
votes
0answers
102 views

fixed points and the cycle index

Let $C_n$ be the cyclic group of order $n$ acting on a finite set $X$ and let $Z(C_n, X; p_1,p_2,\dots)$ be the cycle index of the corresponding permutation group. I wonder whether the knowledge of ...
3
votes
1answer
184 views

Hamiltonian S^1 8-dim manifold with minimal number of fixed points

Let M be a compact symplectic manifold of dimension 8, acted on by S^1, with isolated fixed points, and such that the Betti numbers are the same as the Betti numbers of CP^4. Let "c1" be the first ...
14
votes
1answer
306 views

Is a smooth action of a semi-simple Lie group linearizable near a staionary point?

Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...
4
votes
2answers
400 views

Is there a free action on a given variety?

Given a variety $V$, and a prime $p$ I want to decide if there is a free action of $\mathbb{Z}/p\mathbb{Z}$ on $V$, and to find the generator of an action if it exists. Is there a known algorithm to ...
1
vote
4answers
525 views

Variants of point fixed theorem

Let $E$ be a dual Banach space and $C$ a nonempty convex weak* compact subset of $E$. Let $G$ be a group of weak* continuous linear isometries on $E$. Suppose that $g(C)\subset C$ for all $g\in G$. ...
11
votes
1answer
762 views

Cobounded ⇒ cocompact?

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and $$\operatorname{diam} H/\Gamma\le 1.$$ Is it true that $H/\Gamma$ is compact? Stupid example. Assume the action of ...
5
votes
0answers
250 views

Central extensions of automorphisms of Bruhat-Tits trees

This is the first time I am using Mathoverflow and I am still learning how to use it. That is why I want to begin with a curious question: Does the group of automorphisms of a Bruhat-Tits tree have ...
12
votes
5answers
1k views

A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all, So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$ Is there a general ...
3
votes
0answers
212 views

Finding generalised Lyndon words

Let $\Sigma = \lbrace a_1, \ldots, a_n, A_1, \ldots A_n \rbrace$ (where $A_i = a_i^{-1}$) and $\prec$ be a total ordering on $\Sigma$. Let $\Sigma^*$ be the set of all words (generated by the ...
2
votes
2answers
335 views

Is there an invariant theory explanation of the orbit structure of GL₂ acting on second-diagonal symmetric matrices by g∙X = gXJg^tJ ?

Statement of the Specific Result Let $J$ denote the matrices with ones on the "second diagonal", meaning the diagonal between the (1,n) and (n,1) entry, and zeros elsewhere. So in the case $n=2$, ...
6
votes
1answer
465 views

Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group. It was ...
3
votes
3answers
422 views

Lie group actions and f-relatedness

Background Let $f: M \to N$ be a smooth map between smooth manifolds. Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be $f$-related if for all $p \in M$, $(f_*)_p(X_p) = Y_{f(p)}$; ...
3
votes
2answers
321 views

Orbits of a symplectic group on its Lie algebra in the finite field case

The classical problem regarding the action of symplectic group on its Lie algebra gives rise to the following question in the finite field case. Let $\mathbb F_p$ be a finite field. Then the ...
1
vote
1answer
139 views

A question about iterated quotients in riemannian geometry

Background This can be generalised, but let me be fairly concrete. Let $X$ be a simply-connected riemannian manifold and let $G$ denote the Lie group of isometries, assumed nontrivial. Let $F < ...
1
vote
0answers
454 views

Questions on orbit properties of group action on varieties

Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...