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On manifolds which do not admit (smooth) actions of finite groups

Question: Assume a smooth manifold $M$ does not admit any effective smooth group actions of finite groups $G \neq 1$, does it follow that $M$ also admits no continuous effective group actions of ...
Abenthy's user avatar
  • 517
17 votes
0 answers
580 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, ...
Anton Petrunin's user avatar
9 votes
0 answers
136 views

A self-isometry of the sphere of a strictly convex Banach space that does not move basic vectors

Problem. Let $n\in\mathbb N$, $X$ be a strictly convex $n$-dimensional real Banach space, $S_X=\{x\in X:\|x\|=1\}$ be the unit sphere of $X$, and $e_1,\dots,e_n\in S_X$ be linearly independent points. ...
Taras Banakh's user avatar
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9 votes
0 answers
340 views

Which polynomials in the minors of a matrix are invariant under conjugation?

$\newcommand{\Cof}{\operatorname{cof}}$ This is a cross-post. Let $1<k<n$ be a fixed integer. I am trying to understand "what can be said" about the $k$-degree minors of a linear map $T:V \to ...
Asaf Shachar's user avatar
  • 6,611
8 votes
0 answers
264 views

Generalization of a standard algebraic group theory result for a tensor problem

$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
Ben's user avatar
  • 1,010
8 votes
0 answers
360 views

Is there a 2-categorical, equivariant version of Quillen's Theorem A?

Quillen's Theorem A says that a functor $F:C \to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC \simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ ...
Vidit Nanda's user avatar
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7 votes
0 answers
226 views

Fundamental domains for proper Lie group actions on smooth manifolds

The setting: $M$ an arbitrary smooth manifold, $G$ a Lie group acting effectively and properly on $M$ by diffeomorphisms. Motivation: when trying to figure out the homeomorphism type of the orbit ...
Russ Phelan's user avatar
7 votes
0 answers
136 views

Principal orbits for hamiltonian actions

Let $G$ be a compact Lie group which acts by symplectomorphisms on a symplectic manifold $(M,\omega)$. Futhermore let $\mu \colon M \to \mathfrak g$ be a moment map for this action. Denote by $\Omega =...
Panagiotis Konstantis's user avatar
7 votes
0 answers
293 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 ...
Jim Bryan's user avatar
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6 votes
0 answers
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S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
onefishtwofish's user avatar
6 votes
0 answers
130 views

Mazur-Ulam bases in finite-dimensional Banach spaces

Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
Lviv Scottish Book's user avatar
6 votes
0 answers
126 views

Analysis of solutions to a system of nonlinear ODEs arising from differential geometry

Consider the system of ODEs: \begin{equation} \varphi''\varphi'^{q-1}\psi'^{p-2}=\varphi^{p-1}\psi^{q-1}, \end{equation} \begin{equation} \varphi'^2+\psi'^2=1, \end{equation} where $\varphi>0$, $\...
Yuhang Liu's user avatar
6 votes
0 answers
221 views

A group action on another group action quotient: how to best describe the resulting structure and does it have a name?

Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits. Is there a nice ...
მამუკა ჯიბლაძე's user avatar
6 votes
0 answers
372 views

closed substack of quotient stack

The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
aytio's user avatar
  • 371
6 votes
0 answers
227 views

Recovering SU(2)-space when the orbit space is a 3 sphere with 3 singular orbits

Background: Consider $SU(2)$ action on the 6-dim flag manifold $M=SU(3)/T^2$ via left multiplication. We view $SU(2)$ as a subgroup of $SU(3)$ corresponding to $2\times 2$-block. The action is just ...
Yuhang Liu's user avatar
6 votes
0 answers
137 views

Existence of $G$-map between finite $G$-simplicial complex

Let $X, Y$ be finite free $G$- simplicial complex. What kind of properties are necessary for existence a $G$-map,i.e, a continuous map which preserves $G$-action, from $X$ to $Y$? Does existence of ...
MathFun's user avatar
  • 233
6 votes
0 answers
483 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 ...
Alin Galatan's user avatar
6 votes
0 answers
424 views

Non invertibility of certain integral arising from group action

Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question. Edit 2: I understand from the following post that some part of the previos version of my question has ...
Ali Taghavi's user avatar
6 votes
0 answers
288 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 ...
user36583's user avatar
6 votes
0 answers
285 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 ...
Hadi's user avatar
  • 731
5 votes
0 answers
101 views

Is each Lipschitz action of a finite group on the 3-sphere equivalent to a linear action?

It is known that each action of a compact group on the 2-dimensional $S^2$ sphere is equivalent (=conjugated) to the linear action of a subgroup of $O(3)$ on $S^2$. On the other hand, there exists a ...
Taras Banakh's user avatar
  • 40.8k
5 votes
0 answers
809 views

Quotient of a Lie algebra by a subalgebra - what is it?

Cross-posting from math.SE (asked there 20 days ago). The quotient $G/H$ of a group $G$ by its subgroup $H$ has a $G$-action - every transitive $G$-set is of this form. However, the quotient space $\...
მამუკა ჯიბლაძე's user avatar
5 votes
0 answers
207 views

Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
Ali Taghavi's user avatar
5 votes
0 answers
412 views

Infinity categories with an action of a simplicial group

Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much ...
Harrison Chen's user avatar
5 votes
0 answers
108 views

On finite quotients of unions of $n$ affine varieties

Assume that a finite group $G$ acts on a quasi-projective variety $Q$ (say, over complex numbers) that possesses a Zariski cover by $\le n$ affine varieties. My question is: does the quotient $Q/G$ ...
Mikhail Bondarko's user avatar
5 votes
0 answers
312 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, ...
Alin Galatan's user avatar
5 votes
0 answers
335 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 $[\...
Nico Stammeier's user avatar
4 votes
0 answers
118 views

Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...
Eric Kubischta's user avatar
4 votes
0 answers
219 views

Relative Thom isomorphism

Let $\tilde{X}$ be a space with an action of the symmetric group $\mathfrak{S}_k$ and define $X:=\tilde{X}/\mathfrak{S}_k$ to be the quotient. On the other hand, $\mathfrak{S}_k$ acts on $(\mathbb{R}^...
FKranhold's user avatar
  • 1,623
4 votes
0 answers
128 views

Characteristic classes of quotient manifold

Let $M$ be a compact oriented smooth manifold with boundary and let $G$ be a compact Lie group acting smoothly, orientation-preservingly and freely on $M$. (Under what conditions) is there a ...
Kafka91's user avatar
  • 641
4 votes
0 answers
132 views

Linear vs algebraic unipotent quotient stacks

Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations Type 2: $G$ acts on $\...
Anton Mellit's user avatar
  • 3,572
4 votes
0 answers
113 views

Amalgamated subgroup of an HNN extension finitely generated

Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated. Similarly, by ...
J.L.'s user avatar
  • 321
4 votes
0 answers
103 views

Finite transitive linear subgroups

Let $q$ be a prime power and $d$ an integer. I want to understand the classification of the transitive linear subgroups of $GL_d(\mathbb F_q)$. According to the Wikipedia page https://en.wikipedia.org/...
Ferra's user avatar
  • 509
4 votes
0 answers
899 views

Contracted product of torsors

Given a group $G$ and a left $G$-set $X$, then we can make $X$ a right $G$-set defining the action as $xg:=g^{-1}x$ , or if you prefer we are considering the opposite group $G^{op}$ to make the left ...
HaroldF's user avatar
  • 433
4 votes
0 answers
239 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 $S^{\...
Jens Reinhold's user avatar
4 votes
0 answers
264 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)\::\: t_1,...
user35603's user avatar
  • 411
4 votes
0 answers
147 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. ...
Alejandro's user avatar
  • 940
4 votes
0 answers
267 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 ...
Mark Bell's user avatar
  • 3,125
3 votes
0 answers
95 views

Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...
N. SNANOU's user avatar
  • 383
3 votes
0 answers
77 views

Additional symmetries in Theta-like function

cross-posted from https://math.stackexchange.com/questions/4708694/curious-symmetry-in-a-theta-like-function Let $\Theta : \mathfrak{h}\times \mathfrak{h} \to \mathbb{R}$ be defined as follows $$ \...
Testcase's user avatar
  • 541
3 votes
0 answers
143 views

Semi-direct products and associated graded Lie algebras

Let $G$ and $H$ be groups and $G\ltimes H$ their semi-direct product given by $f\colon G\to \operatorname{Aut}(H)$ satisfying $f(g)=\operatorname{id}_H$ in $H/[H,H]\,$ for all $g\in G$. In this ...
Qwert Otto's user avatar
3 votes
0 answers
65 views

Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
THC's user avatar
  • 4,313
3 votes
0 answers
151 views

A circle action on the E8 manifold

In the paper "A survey of group actions on 4-manifolds" by Allan L. Edmonds on page 5 there is the remark "One should note that the coned-off E8-plumbing manifold admits a circle action ...
Nick L's user avatar
  • 6,933
3 votes
0 answers
84 views

non-smooth manifold with circle action (with fixed points)

I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...
Nick L's user avatar
  • 6,933
3 votes
0 answers
68 views

Coordinates for quasiperiodic motion after reconstruction

Consider a free action of $SO(3)$ on a manifold $M$ and some (reducible) dynamics vector field $X$ on $M$. Suposse that the reduced dynamics $X_{red}$ on $M/SO(3)$ has only fixed points and periodic ...
user2002's user avatar
  • 181
3 votes
0 answers
201 views

How the Galois group acts on a Néron–Severi group of a variety?

Let $K/k$ be a Galois extension with Galois group $\Gamma$ and let $X$ be a variety over $k$. Assume that either $X(k)\neq\varnothing$ or $\mathrm{Br}(k)=0$, the Brauer group of $k$. By the Hochschild-...
Mobius's user avatar
  • 165
3 votes
0 answers
273 views

Is the composition of group quotients a group quotient?

I have two sets $X_1$, $X_2$ each with a corresponding group action $G_1$, $G_2$. Linking the two sets is $f:X_1\to X_2$ that maps orbits of $G_1$ into the same point in $X_2$. In other words $X_1/...
Carles Gelada's user avatar
3 votes
0 answers
178 views

Examples of group actions on statistical manifolds

A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold with a torsion-free affine connection $\nabla$ such that $\nabla g$ is symmetric in all entries. Equivalently, there is a dual affine ...
user2002's user avatar
  • 181
3 votes
0 answers
80 views

Quotients of a fixed manifold by a fixed Lie group

Let $M$ a connected paracompact differentiable manifold. Let $G$ a connected Lie group. I am interested in the possible "regular" (e.g. smooth) quotients of $M$ by actions of $G$. What ...
jpdm's user avatar
  • 141
3 votes
0 answers
314 views

Action on a torus

I was asked the following question: suppose that $M= T^{2n}$ a torus of dimension 2n. And let $\mathbf{Z}/2\mathbf{Z} \subset \mathrm{Homeo(M)}$ such that the space of fixed points $N=M^{\mathbf{Z}/2\...
GSM's user avatar
  • 343