# Questions tagged [group-actions]

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### 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 ...
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### 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 ...
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### 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 $C_n(\mathbb{R}^d)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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-...
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### 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 $X$. ...
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### 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 ...
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### 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. ...
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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: $A^* (G \... 2answers 202 views ### What one group action can tell about another [closed] There are two groups,$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$is mysterious. Perhaps all we know about it ... 0answers 89 views ### Characteristic classes of quotient manifold Let$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$. (Under what conditions) is there a ... 0answers 302 views ### Is there a 2-categorical, equivariant version of Quillen's Theorem A? Quillen's Theorem A says that a functor$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$... 0answers 24 views ### Proof of property for Fiedland entropy I am working with Friedland entropy and there is a proof I cannot figure out how to do. Friedland entropy is defined for$\mathbb{Z}^k$continuos actions$\mathcal{T}$on a topological metric space$X$... 1answer 276 views ### An example in symplectic geometry$\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$the moment map ... 1answer 253 views ### Question about an example in symplectic geometry Let M be a coadjoint orbit of dimension 6 of$SU(3)$, and let T be the maximal torus in$SU(3)$. If we denote$\mu : M \longrightarrow \mathbb{R}^2$the moment map associated to the action of T on M, ... 0answers 105 views ### Equivalence between coactions and actions plus a linearization line bundle Let$G$be an algebraic group over a field$k$, and$\mathbb{P}(V)$is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all ... 1answer 116 views ### What is the orbit of the standard conformal structure on$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})/\...
$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$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 ...