All Questions
Tagged with geometric-group-theory lie-groups
34 questions
16
votes
3
answers
1k
views
How can I tell if a group is linear?
The basic question is in the title, but I am interested in both necessary and sufficient conditions.
I know the Tits' alternative and Malcev's result that finitely generated linear groups are ...
15
votes
1
answer
784
views
The completion of the space of finite groups
Edit: I revise the question based on the comment conversations
Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation.
We define ...
- 356
14
votes
1
answer
340
views
On the homological dimension of a Borel construction
Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$...
- 11.9k
11
votes
2
answers
755
views
Quasi-isometric rigidity of Nil
Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
- 3,749
11
votes
0
answers
679
views
Definition of a uniformly bounded dual of a group
The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
user2412
10
votes
1
answer
706
views
Where to find English translation of Pansu's paper from Ann. Math?
Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
- 28k
8
votes
0
answers
168
views
Do compact universal covers have concentration of measure phenomenon?
$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their ...
- 686
6
votes
1
answer
249
views
Growth function of locally compact groups
Every locally compact second countable group $G$ has a regular left-invariant measure $h$, the Haar measure. On the other hand the Birkhoff–Kakutani Theorem asserts that such groups also admit a ...
5
votes
2
answers
505
views
Is every countable discrete group a subgroup of a non discrete Lie group?
1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group?
2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word ...
- 356
5
votes
1
answer
319
views
"Dimension" of discrete subgroups of infinite covolume in Lie groups
Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact
subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...
- 63.9k
5
votes
0
answers
140
views
Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
- 5,405
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
1
answer
208
views
Connection between degree of growth and return probabilities of random walks on Lie groups
Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...
user47125
4
votes
1
answer
256
views
Action of the isometry group of the hyperbolic 5-space
We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the ...
- 1,692
4
votes
0
answers
136
views
Amenability of the group of outer automorphisms of a connected compact Lie group
Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups.
First, allow me to fix some notations.
Let $G$ ...
- 960
3
votes
2
answers
555
views
Counterexamples to Margulis Normal subgroup theorem in rank 1
Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index.
What are the known counter-examples in rank $1$ ?
I am ...
- 382
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
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
53
views
Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
- 563
3
votes
0
answers
153
views
Estimate word-metric length in free nilpotent groups
I would like to estimate the length of a word in a free nilpotent group. As the first example, I would like to estimate the word metric in the Heisenberg group $H_3$. This is the group of upper ...
- 115
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
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
196
views
Uniform sub-linearity of sub-additive functions on groups
Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...
- 1,101
2
votes
1
answer
203
views
Commensurability classes of subgroups of a nilpotent group
Here is a question I stumbled upon in my research.
Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?
Recall that two ...
- 1,819
2
votes
2
answers
757
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 33
2
votes
0
answers
153
views
Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
- 331
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
124
views
Examples of groups admitting a proper $1$-cocyle for a bounded representation
A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...
- 1,090
1
vote
1
answer
308
views
Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]
As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
- 403
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
0
answers
80
views
Quotient of Euclidean space with maximal volume growth
Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...
- 2,535
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
1
vote
0
answers
189
views
Poincaré inequality for connected Lie groups
Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...
- 527
1
vote
0
answers
134
views
Are lattices in the special real linear group subgroup seperable?
Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : U]...
- 11.3k