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1
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1answer
72 views

Actions of the unit circle on finite complex matrices

Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle. How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on ...
3
votes
1answer
233 views

Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.) Suppose we have an algebraic group $G$ ...
3
votes
1answer
250 views

simultaneous action of GL(n) on the matrices

Consider the action of GL(n,k) on the set MxM where M is the set of all n-by-n matrices over k given by $g.(h,l) \mapsto (ghg^{-1}, glg^t)$. Individually these actions are well studied and good ...
0
votes
0answers
94 views

Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here: Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
4
votes
0answers
182 views

Can Z/2 x Z/2 act freely on an infinite dimensional sphere?

Using that all groups that act freely on some sphere $S^n$ have periodic cohomology, one can see that $\mathbb Z/2 \times \mathbb Z/2$ can not act freely on any $S^n$. But can it act freely on ...
6
votes
1answer
163 views

Free actions of non-amenable groups

Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free ...
5
votes
1answer
116 views

Example of a $G$-sphere that is not a $G$-representation sphere

Let $G$ be a finite group with the discrete topology. To set terminology: a $G$-sphere is a sphere equipped with a continuous $G$-action a $G$-representation sphere is a $G$-sphere obtained from an ...
0
votes
0answers
79 views

a construction on Stiefel manifolds

Are there any references concerning the following space $V(k,N,X)$ and $U(k,N,X)$? And the cohomology of these spaces? Thanks.
0
votes
0answers
67 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 ...
6
votes
2answers
255 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 ...
22
votes
4answers
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 ...
1
vote
1answer
181 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
0answers
57 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
1answer
94 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) = ...
1
vote
2answers
197 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
1answer
155 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
2answers
224 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
0answers
146 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 ...
11
votes
1answer
354 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?
2
votes
0answers
160 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
1answer
153 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
1answer
189 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
1answer
221 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
1answer
106 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
0answers
93 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)} ...
4
votes
1answer
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 ...
5
votes
0answers
201 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
votes
0answers
58 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
0answers
174 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
1answer
323 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 ...
4
votes
0answers
229 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
1answer
158 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
votes
0answers
157 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 $f:S^{n}\to \mathbb{R}^{n}$ ...
8
votes
1answer
320 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 ...
3
votes
0answers
108 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 ...
2
votes
0answers
115 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 $G$. ...
2
votes
1answer
161 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
0answers
123 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 ...
-1
votes
1answer
93 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
0answers
74 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 ...
3
votes
0answers
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 ...
4
votes
0answers
230 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)\::\: ...
6
votes
1answer
177 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 ...
6
votes
1answer
234 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 ...
1
vote
0answers
110 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 = ...
5
votes
1answer
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} ...
5
votes
0answers
193 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 ...
8
votes
4answers
481 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 ...
4
votes
2answers
276 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
1answer
259 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 ...