All Questions
Tagged with lie-groups gt.geometric-topology
101 questions
5
votes
0
answers
590
views
Mal'cev completions of finitely generated torsion-free nilpotent groups
There is some question from geometric group theory:
One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
$\Gamma$ and $\...
4
votes
2
answers
545
views
Is every group object in TopMan a Lie group?
Recall that a Lie group is a group object in the category of C∞ manifolds.
If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure ...
- 54.6k
4
votes
1
answer
419
views
Faithful locally free circle actions on a torus must be free?
Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...
- 385
4
votes
2
answers
267
views
Finite models for torsion-free lattices
Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$?
I know this to be true in many instances (e.g. ...
- 1,255
4
votes
2
answers
363
views
Complexification or 'real'ization of Mapping Class group.
So is there a complexification or 'real'ization of the mapping class group or can it be realised as a lattice in some lie group. like $PSL(2, \mathbb Z)$ in $PSL(2, \mathbb R)$. for g=1 this certainly ...
- 327
4
votes
1
answer
233
views
Flat solvmanifolds?
I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
- 481
4
votes
1
answer
183
views
What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?
Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...
- 143
4
votes
1
answer
230
views
Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi_1(\Sigma) $ of $ \SL_2(\...
- 4,081
4
votes
1
answer
799
views
Transitive action on the sphere
Hello,
One of the subgrouops of $SO(n)$ which acts transitively on the sphere $S^{n-1}$ is the (compact) symplectic group $Sp(n/4)$. The center of $Sp(m)$ is isomorphic to $\mathbb{Z}_2$. Can we embed ...
user30435
4
votes
0
answers
226
views
Is the total space of a $ U_1 $ principal bundle over a compact homogeneous space always itself homogeneous?
Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...
- 4,081
4
votes
0
answers
83
views
Points of failure in definition of X- and A-moduli spaces for arbitrary G
In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...
- 1,131
3
votes
2
answers
298
views
Transitive action on non-orientable $ M $ lifts to orientable double cover
Suppose that $ M $ is non-orientable with transitive action by a Lie group $ G $. Does that imply that some Lie group $ G' $ acts transitively on the orientable double cover $M'$?
This is true for ...
- 4,081
3
votes
1
answer
392
views
Almost free Lie group action
It's known that if a compact Lie group $G$ acts freely on a compact manifold $M$, then the orbit space $M/G$ is a manifold. If we only assume that $G$ acts almost freely (i.e. $G_x$ is finite for any $...
- 307
3
votes
1
answer
263
views
Homogeneous manifold deformation retracts onto compact submanifold
Let $G$ be a connected Lie group. Then by a theorem of Cartan there is a diffeomorphism
$$
G \cong K \times \mathbb{R}^n
$$
where $K$ is a maximal compact subgroup of $G$. Now, let $M$ be a ...
- 4,081
3
votes
1
answer
614
views
Coadjoint orbits and homogeneous symplectic $G$-manifolds
We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...
user21574
3
votes
1
answer
100
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
3
votes
1
answer
425
views
3 dimensional solvmanifolds and Thurston geometries
Does every three dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?
definitions/motivation/background:
A solvmanifold is a manifold $ M $ admitting a transitive action by a ...
- 4,081
3
votes
1
answer
98
views
Locally nilpotent algebraic section of tangent bundle is complete?
Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
user75660
3
votes
0
answers
99
views
Relation of geometric and polyhedral convergence
By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
- 61
3
votes
0
answers
164
views
Equivalent condition for compact group to act transitively
Let $ M $ be a connected manifold. Let $ \pi_1(M) $ be the fundamental group of $ M $.
Suppose there exists a compact group $ K $ that acts transitively on $ M $. Then $ \pi_1(M) $ must have a finite ...
- 4,081
3
votes
0
answers
159
views
Convergence of Fuchsian groups and existence of suitable homeomorphisms
Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
3
votes
0
answers
402
views
Is every manifold a double coset space?
Given any manifold $ M $ does there exist $ G $ a Lie group and $ H,\Gamma $ closed subgroups of $ G $ such that
$$
M \cong \Gamma \backslash G/H
$$
I was inspired to ask by this question: Example ...
- 4,081
3
votes
0
answers
267
views
What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?
What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?
Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...
- 679
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
3
votes
0
answers
888
views
Quotient space, homogeneous space, and higher homotopy groups
Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...
- 10.5k
2
votes
1
answer
719
views
Classification of isometries of hyperbolic 3-space
Denote the upper half space by $\mathcal{H}_{3}=\Bbb{C}\times (0,\infty)$. A point $P \in \mathcal{H}_{3}$ is given as, $P=(z, t)=(x, y, t)=z+t j$ where $z=x+i y$ and $j=(0,0,1) .$ The group $P S L_{2}...
- 33
2
votes
1
answer
311
views
Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?
Let $G$ be a connected compact Lie group, consider the left/right action on itself.
For any finite $A\subset G$, consider the centralizer
$Z_G(A):=\{g\in G| a g= g a\}$.
Q: is $Z_G(A)$ a connected ...
- 21
2
votes
1
answer
484
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 4,081
2
votes
2
answers
213
views
Riemannian homogeneous equivalent to linear group orbit
Let $ M $ be a smooth manifold.
Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $.
Recall that a manifold $ M $ is Riemannian homogeneous if ...
- 4,081
2
votes
1
answer
133
views
Do discrete embeddings of surface groups not necessarily carry an embedding of SL_2?
We can get arithmetic lattices isomorphic to free groups in $\mathrm{SL}_2\mathbb{R}$, so in general we can’t expect homomorphisms of lattices into semisimple Lie groups to say much about $\mathrm{SL}...
- 779
2
votes
1
answer
251
views
Basics on lattice in classical groups
as a beginner,I am not sure whether this question is too basic to post here./-\。
Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...
- 21
2
votes
1
answer
138
views
noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous
Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
- 4,081
2
votes
0
answers
76
views
Weakening of orbit type stratification to non-proper Lie group actions
I am working in a setting where I am thinking about effective, fixed-point free actions of $\text{PSL}(2; \mathbb{R})$ on compact three manifolds by homeomorphisms. I'm not sure if this is relevant ...
- 21
2
votes
0
answers
108
views
Questions about symmetric spaces
I'm a little confused with the following questions:
(1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$?
(2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
- 563
2
votes
0
answers
83
views
A quasi-isometric embedding of a convex cocompact subgroup
I am currently reading a paper where they state the following claim:
"For a convex cocompact representation $\rho: \Gamma \to G$, the existence of a cocompact invariant convex set $\mathcal{C}$ ...
- 123
2
votes
0
answers
108
views
Cohomology of colored braid groupoids
Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
- 2,533
2
votes
0
answers
127
views
Homeomorphism type of the horofunction boundary for nilpotent Lie groups
Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form
$$\beta_y(x)=d(x,y)-d(w,y).$$
The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...
- 2,090
1
vote
1
answer
301
views
Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure
I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...
user21574
1
vote
1
answer
130
views
Structures on open surfaces
Let $\phi\in PSL(2,R)$ be hyperbolic and $\varphi\in PSL(2,R)$ be elliptic.
Is it possible to find a local homeomorphism $f:H^2\rightarrow H^2$ such that
$f(\phi(x))=\varphi(f(x))$ for all $x\in H^2$ ?...
- 31
1
vote
1
answer
192
views
Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
- 447
1
vote
2
answers
661
views
module of sections of the horizontal bundle
Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth ...
- 1,222
1
vote
1
answer
182
views
A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
- 331
1
vote
1
answer
214
views
Orbits of Product Lie Groups Action
Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
user30435
1
vote
0
answers
96
views
Constructing homeomorphisms from continuous functions and matrix exponentials
Fix a $d\times d$ matrix $A$, let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a continuous function, and define the induced map $F_{f,A}:\mathbb{R}^d\rightarrow \mathbb{R}^d$ by
$$
x \mapsto \exp(f(x)A)...
- 5,405
1
vote
0
answers
137
views
When can a spatial Manifold be deformed into another spatial manifold?
I'm reading a paper by Gaiotto, Kapustin, Seiberg, and Willett, titled "Generalised Global Symmetries" (MSN). In section 3.1, paragraph 4 they write something to the order of:
If we have a $M^d$ ...
- 569
1
vote
0
answers
254
views
Defining a notion of “volume of its lattice” for non-rational subspaces
Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...
- 3,079
1
vote
0
answers
180
views
Cocompact (finite covolume) lattices in euclidean groups
1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)?
2) Also what is (if any) the ...
- 11
1
vote
0
answers
83
views
Rigidity of lower-dimensional lattices in Euclidean groups
Informal intro / motivation:
Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer ...
- 13.6k
0
votes
1
answer
296
views
Compact Lie groups with only 3 dimensional cohomology generators
Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.
For which $M$, $...
0
votes
0
answers
172
views
Generating Set for $O(V)$ over $\mathbb Z_2$
I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
- 105