Questions tagged [group-actions]

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3
votes
1answer
179 views

Trans-amenability of group actions

This problem is derived from this post. Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are ...
1
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0answers
98 views

Fixed point subalgebra

Suppose that $M$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^{G}$ the fixed point subalgebra and suppose that $M^{G}=\mathbb{C}$ (i.e., we have an ergodic ...
1
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0answers
22 views

On the higher-dimensional Berry-Robbins problem

Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...
3
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0answers
55 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 ...
7
votes
1answer
194 views

Is $\mathbb{Q}$ "equivalent" to a structure with transitive automorphism group action?

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For ...
1
vote
1answer
91 views

connected components of the fixed point subvariety

Let $X$ be a smooth complex variety with an action of a finite group $G$. The fixed point subvariety $X^G$ is smooth but may have many connected components. What determines these connected components ...
8
votes
1answer
252 views

Scalar curvature and the degree of symmetry

Let $M$ be a closed connected smooth manifold, then we define the degree of symmetry of $M$ by $N(M):=\sup_\limits{g}\{\mathrm{dim}(\mathrm{Isom}(M,g)\}$, where $g$ is a smooth Riemannian metric on ...
3
votes
1answer
133 views

Equivalent definitions of strongly proximal action

Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar, Kennedy and Ozawa: I have two questions: (1) What ...
2
votes
1answer
148 views

Behavior of canonical divisor under a finite group quotient

Given a smooth algebraic surface $X$, and a group $G$ acting on it and letting $Y := X / G$, how can we compute $K_Y^2$ from from $K_X^2$? Current progress: In Borisov and Fatighenti - New explicit ...
5
votes
2answers
236 views

Sections of a polar action are totally geodesic

This question is a repost of the following: https://math.stackexchange.com/questions/4195805/sections-of-a-polar-action-are-totally-geodesic. I've decided it to post it here because it didn't seem to ...
1
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0answers
50 views

Row-wise conjugation of completely bounded map by group action

Let $B$ be a $G$-$C^*$-algebra and let $\phi\colon B \to B$ be a completely bounded map (not necessarily $G$-equivariant). For group elements $F := \{h_1, \ldots, h_k\} \subset G$ we consider the ...
3
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0answers
59 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 ...
2
votes
0answers
131 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-...
3
votes
1answer
138 views

Maximal closed subscheme stable under the action of a finite connected group scheme

Let $k$ be a field of characteristic $p>0$, $X$ a smooth projective $k$-variety and $Y\subseteq X$ a closed irreducible subvariety. Let $G$ be a connected finite $k$-group scheme acting on $X$. ...
3
votes
0answers
240 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/...
1
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0answers
89 views

Quotient measure on locally compact spaces

Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \...
3
votes
2answers
216 views

Three dimensional real Lie groups with cocompact discrete subgroups

I would like to know what are all the real three dimensional Lie groups (simply connected) that can act transitively and locally freely on a compact three dimensional manifold? This is equivalent to ...
7
votes
1answer
204 views

How special are homogeneous spaces?

Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$? Related questions/approaches: Of course we need $\...
2
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0answers
85 views

Is this equivariant function constant?

Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
7
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0answers
251 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 ...
6
votes
0answers
103 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 ...
5
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0answers
82 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 ...
9
votes
0answers
120 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. ...
1
vote
1answer
142 views

Statistical manifolds with trivial statistical structure after quotienting

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 ...
3
votes
0answers
134 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 ...
1
vote
1answer
103 views

Is the union of two proper flows proper?

Let $\varphi _t$ be a flow, aka. a one parameter group of homeomorphisms of the open subset $\Omega \subseteq {\mathbb R}^n$, which we assume to be continuous in the usual sense that $$ (t, x)\in {\...
1
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0answers
70 views

About irreducible connection

The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...
4
votes
1answer
105 views

A combinatorial characterization of the central inversion of a polytope?

Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...
2
votes
0answers
113 views

Theorem classifying fixed point sets of an isometry of the three sphere

Let $T:S^3\rightarrow S^3$ be an isometry of finite order. Then the set $S^T=\{x\in S^3|Tx=x\}$ of fixed points is either empty, or a pair of antipodal points, or a great circle, or a great sphere. ...
4
votes
0answers
127 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}^...
2
votes
1answer
182 views

Group action: 'Minimal' subgroup generating an orbit

Let $G$ be a symmetric group on a finite set acting on another finite set $X$ through a natural action $\alpha:G \times X \to X$, $\alpha(g,x)=gx$. Let $x \in X$ and consider the orbit $G \cdot x := \{...
2
votes
1answer
169 views

Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold

I'm reading the article of equivariant cohomology with generalized coefficients by Kumar and Vergne and I have this question from that article Let G be a compact lie group with lie algebra $\mathfrak{...
4
votes
1answer
108 views

Does a periodic and strongly invertible knot have a specific type of diagram?

Let $K \subset S^3$ be a knot which is both strongly invertible and periodic, that is, $K$ is fixed by both a smooth involution $\tau: S^3 \to S^3$ which preserves the orientation of $S^3$ but ...
4
votes
1answer
169 views

Locally trivializing a G vector bundle?

In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
3
votes
0answers
70 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 ...
6
votes
1answer
111 views

Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$

In Curvature and symmetry of Milnor spheres, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper,...
3
votes
0answers
251 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\...
1
vote
0answers
46 views

Identifications of the space $A^* (G \times _H M) $

Let $G$ be a lie group, and let H be lie subgroup of G acting on G by right translation. Let M be a H-manifold. Why do these equalities hold: For the algebra $ A^* (G \times _H M)$, we have: $ A^* (G \...
3
votes
2answers
202 views

What one group action can tell about another [closed]

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$. $S$ may have some structure. The groups may too. The actions respect them. $G_1$ is mysterious. Perhaps all we know about it ...
4
votes
0answers
89 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 ...
8
votes
0answers
302 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 $F/d$ ...
0
votes
0answers
24 views

Proof of property for Fiedland entropy

I am working with Friedland entropy and there is a proof I cannot figure out how to do. Friedland entropy is defined for $\mathbb{Z}^k$ continuos actions $\mathcal{T}$ on a topological metric space $X$...
3
votes
1answer
276 views

An example in symplectic geometry

$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...
6
votes
1answer
253 views

Question about an example in symplectic geometry

Let M be a coadjoint orbit of dimension 6 of $SU(3)$, and let T be the maximal torus in $SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map associated to the action of T on M, ...
0
votes
0answers
105 views

Equivalence between coactions and actions plus a linearization line bundle

Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ...
3
votes
1answer
116 views

What is the orbit of the standard conformal structure on $S^2$ under $\operatorname{SL}(3,\mathbb{R})$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$Consider the group $\GL_+(3,\mathbb{R})$ acting on $\mathbb{R}^3$. It induces an action of $\GL_+(3,\mathbb{R})/\...
9
votes
1answer
171 views

Almost free circle actions on spheres

$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem: Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then ...
0
votes
1answer
66 views

Is there a $G$-paradoxical $G$-invariant subset of the plane for $G$ a group of rigid motions?

The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $...
0
votes
2answers
127 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
6
votes
1answer
274 views

Group action with unique word

This must be known or easy for some of you, but here goes: Suppose $f_0,f_1:[n]\to [n]$ are invertible functions, where $[n]=\{0,\dots,n-1\}$ is a set of $n$ elements. For a word $w=w_1\dots w_m\in\{...

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