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4
votes
1answer
26 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 ...
15
votes
0answers
399 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, ...
7
votes
0answers
114 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 ...
6
votes
0answers
180 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 ...
5
votes
0answers
174 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 ...
5
votes
0answers
254 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 ...
4
votes
0answers
163 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, ...
4
votes
0answers
191 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 ...
4
votes
0answers
115 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 ...
4
votes
0answers
219 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)\::\: ...
4
votes
0answers
134 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 ...
3
votes
0answers
110 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 ...
3
votes
0answers
85 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 ...
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. ...
3
votes
0answers
500 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 ...
3
votes
0answers
220 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
0answers
146 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
votes
0answers
99 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
129 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 ...
2
votes
0answers
78 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 ...
2
votes
0answers
206 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 ...
1
vote
0answers
149 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 ...
1
vote
0answers
112 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 $G$. ...
1
vote
0answers
131 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 ...
1
vote
0answers
78 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 ...
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 ...
1
vote
0answers
172 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 ...
1
vote
0answers
463 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 ...
0
votes
0answers
66 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)} ...
0
votes
0answers
38 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 ...
0
votes
0answers
72 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 ...
0
votes
0answers
98 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 = ...
0
votes
0answers
118 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 ...
0
votes
0answers
88 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 ...
0
votes
0answers
107 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 ...