Questions tagged [group-actions]

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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 &...
Rebecca J. Stones's user avatar
9 votes
3 answers
1k views

Relation between groups and classifying spaces

Let $G$ be a nonabelian group, with classifying space $BG$. Motivation: We can compute its homology, $H_\ast(BG)=H_\ast(G)$. It would be nice to see some equivariant computations, like $H_\ast^G(BG)$ ...
Chris Gerig's user avatar
  • 17.1k
9 votes
3 answers
2k views

Lie group actions and f-relatedness

Background Let $f: M \to N$ be a smooth map between smooth manifolds. Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be $f$-related if for all $p \in M$, $(f_*)_p(X_p) = Y_{f(p)}$; ...
José Figueroa-O'Farrill's user avatar
9 votes
2 answers
2k views

Which weighted projective spaces (and their finite quotients) are local complete intersections?

Let $G$ be a finite subgroup of $\textrm{Gl}_{n+1}(k)$ (where $k$ is an algebraically closed field). My question is: do there exist examples of $G$ such that the corresponding quotient $P$ of $\mathbb{...
Mikhail Bondarko's user avatar
9 votes
2 answers
623 views

Action of the homotopy braid groups on reduced free groups

Firstly some definitions: $B_n$ is the braid group with $n$ strands. $\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...
Zuriel's user avatar
  • 1,098
9 votes
1 answer
912 views

Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...
Lisa's user avatar
  • 93
9 votes
2 answers
3k views

Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
user33576's user avatar
9 votes
1 answer
302 views

Counterexamples to analogue of Cannon conjecture in higher dimensions

It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$. The analogous statement for $\mathbb{H}^...
user68316's user avatar
  • 245
9 votes
2 answers
367 views

Is there a highly transitive action of a finitely generated torsion simple group?

Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ? Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two $k$-...
Pablo's user avatar
  • 11.2k
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 ...
ofiz's user avatar
  • 607
9 votes
4 answers
1k views

Action of a Lie group with finitely many orbits

EDIT: Let a real Lie group $G$ act on a smooth manifold $M$ with finitely many orbits such that each orbit is locally closed ($M$, but not $G$, may be assumed to be compact in my case). Let $\mathcal{...
user51305's user avatar
9 votes
1 answer
803 views

6-manifolds admitting SO(3) action with 2 orbit types

Let $M^6$ be a 6-dimensional smooth manifold, on which the group $G=SO(3)$ acts smoothly with 2 orbit types $SO(3)/SO(2)$ and $SO(3)$, such that the orbit space $X=M/SO(3)$ is a 3-ball $B^3$, whose ...
Yuhang Liu's user avatar
9 votes
1 answer
213 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 ...
CuriousUser's user avatar
  • 1,420
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(...
Caramello's user avatar
  • 382
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
  • 40.8k
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
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 \...
Mostafa's user avatar
  • 403
8 votes
5 answers
4k views

Group Action with a Fixed-Point Property

Is there an example of an infinite group $G$ acting on a set $X$ such that each non-identity element of $G$ fixes exactly two elements of $X$, and some two elements of $G$ have in common exactly one ...
Martin Erickson's user avatar
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 ...
ggt001's user avatar
  • 141
8 votes
2 answers
476 views

What does the free action of a surface group on an R-tree look like?

Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, ...
user32157's user avatar
  • 337
8 votes
2 answers
406 views

Fixed point of $S^1$-action using roots of unity

Fact: For any (continuous) $S^1$-action on the closed unit disk $\mathbb{D}^n$, there is a fixed point $x_0\in\mathbb{D}^n$. I have thought of a possible argument that re-proves this, but am not sure ...
Chris Gerig's user avatar
  • 17.1k
8 votes
1 answer
3k views

Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...
Jon's user avatar
  • 83
8 votes
2 answers
868 views

Unipotent algebraic group action on quasi-affine (vs. affine) variety?

This question arises from a comment by user nfdc23 on an unrelated recent MO question here. It concerns textbook treatments of what has been called the "Theorem of Kostant-Rosenlicht", stated as ...
Jim Humphreys's user avatar
8 votes
1 answer
274 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 $\...
GFR's user avatar
  • 639
8 votes
1 answer
720 views

Status of Hilbert-Smith conjecture and H-S conjecture for Hölder actions

The Hilbert-Smith conjecture states that If $G$ is a locally compact group which acts effectively on a connected manifold as a topological transformation group then is $G$ a Lie group. It was ...
Zarathustra's user avatar
  • 1,404
8 votes
2 answers
412 views

Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite? Consider the ...
Lawrence Mouillé's user avatar
8 votes
1 answer
652 views

Center of one-point stabilizer in 2-transitive groups

In this MO question it was mentioned that the following fact seems to be true: If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a non-trivial center, then $G$ is of ...
Leandro Vendramin's user avatar
8 votes
1 answer
300 views

What is this concept related to group actions?

Inspired by Group Equivariant Convolutional Networks by Taco Cohen and Max Welling I have been thinking about the following construction and I wonder where else it has been described and studied. ...
Tom Ellis's user avatar
  • 2,775
8 votes
1 answer
550 views

Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...
John Pardon's user avatar
  • 18.3k
8 votes
1 answer
276 views

Groups that act transitively on $\mathrm{Gr}(k,\Bbb R^n)$ but not transitively on $\mathrm{Gr}(k+1,\Bbb R^n)$

Is it known for which $n, k\in\Bbb N$ there exists a matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^n)$ that acts transitively on $\mathrm{Gr}(k,n)$, i.e., on the $k$-dimensional subspaces of $\Bbb ...
M. Winter's user avatar
  • 12.5k
8 votes
1 answer
355 views

Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones

Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but ...
user avatar
8 votes
1 answer
250 views

$\mathbb{C}^{*}$-actions on Fano $3$-folds

I am looking for an example of a smooth Fano $3$-fold $X$ over $\mathbb{C}$, with a non-trival $\mathbb{C}^{*}$-action, which satisfies the following properties: There is a $\mathbb{C}^{*}$-action ...
Nick L's user avatar
  • 6,923
8 votes
1 answer
429 views

Free group actions on varieties and algebras of coinvariants

Suppose $k$ is an algebraically closed field of characteristic zero and $A$ is a finitely generated commutative associative reduced $k$-algebra. Suppose the group $\mathbb{Z}_2$ acts on $A$ in such a ...
Alistair Savage's user avatar
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
  • 15.4k
7 votes
2 answers
865 views

Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true: For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections. ( ...
Nico Bellic's user avatar
7 votes
1 answer
415 views

Free $\mathbb{Z}_2$-actions match at some point

I have in front of me a proof of this lemma: If $f$ and $g$ are free $\mathbb{Z}_2$-actions on $S^1$, then $f(x)=g(x)$ for some $x \in S^1$. A $\mathbb{Z}_2$-action on the unit circle $S^1$ is a ...
Joseph O'Rourke's user avatar
7 votes
1 answer
341 views

On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$. Is it true ...
asv's user avatar
  • 21.1k
7 votes
2 answers
500 views

Algebraic proof without using comparison theorem for étale cohomology

Let $X$ be some smooth scheme over $\mathbf C$ equipped with an action of $\mu_n$ (the group of $n$th roots of unity). The étale cohomology groups of X are therefore equipped with an action of $\mu_n$...
Oblomov's user avatar
  • 2,501
7 votes
1 answer
423 views

Understanding a germ of a GIT quotient

Let $X$ be a smooth complex affine variety, let $G$ be a complex reductive group acting on $X$. Suppose that the stabilizer $G_x$ of a point $x\in X$ is reductive and connected. Let $\varphi: X\to X//...
aglearner's user avatar
  • 14k
7 votes
1 answer
582 views

Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
THC's user avatar
  • 4,313
7 votes
1 answer
414 views

How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?

This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map $$ \mu: G\times X\...
Zhaoting Wei's user avatar
  • 8,657
7 votes
1 answer
262 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 ...
Noah Schweber's user avatar
7 votes
2 answers
120 views

Is there a subset of $\Sigma_n$ s.t. each pair of elements is once in each pair of positions?

Is there a subset $A \subset \Sigma_n$ such that for each pair $(x, y)$ and each pair $(i, j)$, there is exactly one permutation $\sigma \in A$ such that $\sigma(i) = x$ and $\sigma(j) = y$? Remark ...
user91088's user avatar
7 votes
1 answer
299 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
Dylan Thurston's user avatar
7 votes
1 answer
273 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, ...
Maria's user avatar
  • 133
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 $...
Paolo Piccione's user avatar
7 votes
1 answer
899 views

When is a conjugacy class of matrices an embedded submanifold?

Let $M_{n\times n}$ denote the set of $n\times n$ real matrices and let $GL_n$ be the subgroup of invertible matrices. $GL_n$ acts on $M_{n\times n}$ smoothly by conjugation, which means that each ...
Eric O. Korman's user avatar
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 $\mathscr{B}$...
Matteo Bruno's user avatar
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

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