Questions tagged [group-actions]

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1 vote
0 answers
66 views

Expressing the union of principal orbits as a disjoint union of global slices for proper group actions

Setup: I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes. Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
0 votes
0 answers
40 views

Orbits/affine spaces in GAP

Another GAP-related question. I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...
1 vote
0 answers
83 views

Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
1 vote
0 answers
61 views

Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$. Let us consider a Schwartz space $\mathcal{S}$ defined as \begin{equation} \mathcal{S}:= \Bigl\{ \...
6 votes
1 answer
142 views

References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
2 votes
0 answers
42 views

Topology of an orbit space constructed from a Fréchet space under the "local" action of some "smooth" group

Let $G$ be a nontrivial connected compact subgroup of the general linear group $\operatorname{GL}(\mathbb{R}^3)$. For example, we may take $G$ to be $\operatorname{SO}(3)$. Next, let $\mathcal{S}(\...
2 votes
0 answers
228 views

Action of algebraic group in cohomology of equivariant algebraic vector bundle

Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
4 votes
1 answer
162 views

Explicit formula for general group extension in terms of cartesian product set

According to Wikipedia and ncat lab general group extensions $$N\rightarrow G\rightarrow Q$$ are classified by a group homomorphism $$\rho: Q\rightarrow \operatorname{Out}(N)$$ subject to a constraint ...
4 votes
1 answer
198 views

If a discrete and faithful representation of a surface group has proximal values, does the attracting points map have a continuous extension?

For some context, I'm studying the paper Anosov Representations and Proper Actions [GGKW]. $G$ denotes a non-compact real reductive Lie group of rank greater than $1$, $\Gamma$ denotes the fundamental ...
1 vote
0 answers
92 views

Almost simple groups and their involutions without CFSG

Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...
0 votes
0 answers
164 views

Research directions related to the Hilbert-Smith conjecture

The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
9 votes
2 answers
510 views

Quotients of schemes by connected groups

Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$. By the Keel-Mori theorem, the quotient $X/G$ is ...
2 votes
0 answers
59 views

Name for a product of actions / dynamical systems

Suppose $G \curvearrowright X, H \curvearrowright Y$ are group (or monoid) actions, or dynamical systems. Then $X \times Y$ is a $G \times H$-system of the same type in the obvious way by $(g, h) \...
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 ...
0 votes
1 answer
186 views

The description of Hurwitz groups

Let $G$ be a Hurwitz group, i.e. the automorphism group of some Hurwitz surface $C$. Then Hurwitz's automorphisms theorem shows that the quotient map of $C$ by $G$ has ramification points of indices $...
1 vote
0 answers
114 views

Smooth action on cotangent space of the plane

Assuming that $S^1$ acts on $\mathbb{R}^2$ by smooth maps (which are diffeomorphisms), the induced action on the cotangent bundle given by $$g\cdot(x,\xi)=(g\cdot x,\varphi^∗_{g^{−1}}\xi)$$ acts via ...
7 votes
1 answer
314 views

volume of exceptional group orbits

Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by $...
7 votes
1 answer
164 views

Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle. Considering the moduli space of connections $\mathscr{B}$...
0 votes
0 answers
42 views

Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on

For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
8 votes
5 answers
2k views

Examples of manifolds with effective circle actions?

I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d \...
4 votes
1 answer
138 views

Examples of discrete quantum group actions on commutative von Neumann algebras

Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ ...
1 vote
0 answers
113 views

A compact Lie group $G$ acting on a compact Lie group $K$ transitively. Is there a $C$ such that $d(gx,gy)\leq Cd(x,y)$?

Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $...
4 votes
1 answer
345 views

Action of braid groups on regular trees

Question: Are there any well known actions of braid groups on trees? For example is there some action of a braid group $ B_n $ on a $ p $ regular tree for some $ p $ such that the action is transitive ...
1 vote
1 answer
99 views

Nonamenable p.m.p. action on a standard probability space

Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations. Is the action of $G$ always amenable? (Amenable action, ...
4 votes
1 answer
145 views

An analogue of Mostow-Palais equivariant embedding theorem for the group of conformal automorphisms of the 2-sphere

Is there a smooth embedding of $S^2$ into some Euclidean space that is equivariant with respect to a linear representation of $PSL(2,\mathbb C)$? A counterexample to a more general question can be ...
0 votes
1 answer
47 views

Holomorphic cyclic action on smooth toric manifold extends to C^* action?

Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?
0 votes
1 answer
105 views

Integral mean value property

Let $V$ be the space of all continuous functions $f$ on the real line with $f(x)=\frac12\big(f(x-1)+f(x+1)\big)$. It contains the space of periodic functions. The latter equals the space of ...
2 votes
0 answers
76 views

Bialynicki-Birula decomposition for $\mathbb{G}_m$-actions on projective schemes

The classical BB-decomposition works for non-singular projective varieties. Here I want to consider projective schemes, in particular when the scheme is not reduced. Let $\Bbbk=\mathbb{C}$. Let $X$ be ...
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 $$ \...
6 votes
0 answers
132 views

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 ...
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 ...
1 vote
0 answers
144 views

Groups acting on categories produce 2-cocycles

$\DeclareMathOperator\Hom{Hom}\newcommand\id{\mathrm{id}}\DeclareMathOperator\Aut{Aut}$Let $\mathcal{C}$ be a category (such that each hom sets are $\mathbb{C}$ linear spaces) and $G$ be a group. We ...
2 votes
0 answers
61 views

Lifting paths along group quotients relative to a base

Suppose you have a map of topological spaces $X\to S$, an $S$-group $G\to S$ (i.e. a group object in $\mathrm{Top}_{/S}$), an action of $G$ on $X$ relative to $S$ which is free and properly ...
8 votes
1 answer
420 views

dichotomy in hyperbolic groups

Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...
0 votes
0 answers
57 views

How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?

Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...
0 votes
0 answers
75 views

Examples of amenable, Hausdorff, locally compact, second countable groups which are not discrete, not compact, and not abelian

I'm working on a problem that involves an amenable group acting on some set by bijections. Initially, I assumed the group was discrete and the set was countable, however I realized that the arguments ...
4 votes
1 answer
408 views

Group action on a condensed set and its orbit space

Let $X$ be a condensed set, and let $G$ be a (discrete) group. Suppose we have an action $G$ on $X$, which is a group morphism $a:G \rightarrow \mathrm{Aut}(X)$, where $\mathrm{Aut}(X)$ is the group ...
1 vote
1 answer
387 views

Does the semi-stable set determine the linearization of a GIT quotient?

Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group. Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the ...
2 votes
0 answers
45 views

Lifting a group action to a Banach bundle

I have been searching the literature for results on lifting a group action from the base space of a Banach bundle (Important note: NOT Banach VECTOR bundle). The setting I am interested in weaker than ...
0 votes
0 answers
76 views

Methods for calculating (one-parameter subgroup) actions

For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form \begin{equation} \mathrm{e}^{t L(z)} f(z) \end{equation} ...
2 votes
1 answer
214 views

Projective G-group

Let $G$ be a fixed group. By definition, a $G$-group is a group $X$ with a $G$-action that respects the group operation of $X$. A free $G$-group means a group freely generated by a free $G$-set. A &...
2 votes
0 answers
279 views

A possible invariant associated to a compact group

Let $G$ be a compact topological group with normalized Haar measure $\mu$. Is there an effective isometric action of $G$ on some $\mathbb{R}^n$ such that the following map would be a non-irreducible ...
3 votes
1 answer
93 views

Do we have an equivariant version of integrability theorem of flat connections?

I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1: Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...
0 votes
0 answers
164 views

Why Lubin Tate character acts on torsion points of CM elliptic curve implies the group of torsion points is infinite?

Let $F$ be quadratic imaginary field, and $R_F$ be its ring of integers. Let $E /\Bbb{Q} $ be an elliptic curve which has CM by $F$. Suppose $E$ has good reduction at $P$,which is prime ideal of $R_F$....
3 votes
1 answer
112 views

Unitary in adjointable operators associated with equivariant Hilbert module

Consider the following fragment from the article "Tannaka–Krein duality for compact quantum homogeneous spaces. I. General theory" by De Commer and Yamashita: How exactly is $\mathcal{E}\...
2 votes
0 answers
51 views

Different invariants of group actions from isomorphic subgroups

Consider $D_8,$ the dihedral group of order $8$, acting on the unit square $X=[0,1]^2 \subseteq \mathbb{R}^2$ in the natural way– essentially take the unique linear extension of the action on the ...
9 votes
1 answer
478 views

Is it possible to average a riemannian metric over an action and preserve curvature bounds?

Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric $$g'(X,Y)=\int_Gg(...
11 votes
2 answers
891 views

Not very transitive actions

Suppose $m$ is a positive integer. I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-...
6 votes
2 answers
346 views

The action of a subgroup of the torsion group of elliptic curves on integral points?

Let $E$ be an elliptic curve given in long Weierstraß form with all coefficients $a_1,a_2,a_3,a_4,a_6 \in \mathbb{Z}$. It is known that the rational points $E(\mathbb{Q})$ form a group which has a ...
1 vote
0 answers
42 views

What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?

$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...

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