All Questions
Tagged with gr.group-theory ds.dynamical-systems
29 questions with no upvoted or accepted answers
15
votes
0
answers
477
views
Diffeomorphisms of $\mathbf R^n$
Let $G={\rm Diff}_0^c(\mathbf R^n)$, $n\geq 1$, be the group of compactly supported diffeomorphisms isotopic to the identity through compactly supported isotopies.
Question: Is there an example to ...
- 1,782
12
votes
0
answers
268
views
If two group actions lead to the same orbifold, are they conjugate?
In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...
- 616
11
votes
0
answers
258
views
Minimal actions commuting with amenable actions of $\mathbb{F}_2$
For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
- 211
11
votes
0
answers
203
views
Fundamental groups of reduced subgroup lattices
Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
- 15.5k
10
votes
0
answers
466
views
For a group with one end does the property of connected spheres follow?
One of my friends is studying group actions on the circle, and he ended up with a question in geometrical group theory. Let us consider a finitely generated group $G$ with generators $g_1, \ldots g_n$....
- 1,143
7
votes
0
answers
286
views
Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?
Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
user35370
7
votes
0
answers
305
views
Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action
Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
- 2,560
5
votes
0
answers
203
views
Terminology question in group actions
Given a continuous group action $G \times X \rightarrow X$ on a topological space $X$, is there a standard term for the subsets $K \subset X$ for which
Every open neighborhood of $K$ intersects every ...
- 13.5k
5
votes
0
answers
210
views
Divisible orientation preserving diffeomorphism which is time-$1$ map of no smooth flow
Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that
for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of ...
- 356
5
votes
0
answers
135
views
Possible homogeneity of infinite dimensional Sierpinski carpet analogues?
Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion.
Construct subsets $S_n$ by removing points from $H$ if for any $m$,
at least $n$ of the ...
- 17.6k
5
votes
0
answers
352
views
"topological" conjugacy of group automorphisms
In the paper "Orbit Equivalence and Topological Conjugacy of Affine
Actions on Compact Abelian Groups", S. Bhattacharya shows (Theorem 3) the following:
Theorem. Given two actions $\alpha$ and $\...
- 3,897
4
votes
0
answers
108
views
Decidability of whether two polynomial bijections generate a free group
I am wondering about the decidability of the following question:
Given two polynomial bijections $f, g$ from the real numbers to the real numbers (with say rational coefficient just to simplify what &...
- 1,075
4
votes
0
answers
327
views
Amenable groups acting on the real line, that are not subexponentially-amenable
In the literature, there are several examples of solvable groups acting faithfully by order-preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth with ...
3
votes
0
answers
115
views
Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 101
3
votes
0
answers
76
views
Embedding powers of a subshift into full shifts
Let $X$ be a subshift over a finitely generated group $G$.
Let $K(X)$ denote the smallest cardinality of any alphabet $A$ such that $X$ embeds (continuously and $G$-equivariantly) into $A^G$. For ...
- 611
3
votes
0
answers
142
views
"Spectral decomposition" action on the unitary group
Consider a matrix $U$ from the unitary group $U_N(\mathbb{C})$ and consider the map $f:U_N(\mathbb{C})\rightarrow U_N(\mathbb{C})$ where $f(U)$ is the matrix of the eigenvectors of $U$.
What is ...
- 2,135
3
votes
0
answers
309
views
A Dedekind Eta trajectory / horocyclic flow (Reference request)
I've been exploring the composition of essentially the Dedekind $\eta$-function with
parabolic Möbius transformations,
$$C_L(z,t)=\left(\frac{z}{-tz+1}\right)^{\frac{1}{2}}\eta\left(\frac{z}{-tz+1}\...
- 10.5k
3
votes
0
answers
148
views
Actions of the discrete Heisenberg group by formal power series of two variables
I am interested in faithful actions of the discrete Heisenberg group $H$ by smooth diffeomorphisms of a surface $S$, that is, 1-1 homomorphisms $\phi \colon H \to \text{Diff}^{\infty}(S)$.
We say $p \...
- 616
2
votes
0
answers
153
views
Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
- 331
2
votes
0
answers
158
views
Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$
I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as
$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\cdot \big(v,\,w\big)\...
- 203
2
votes
0
answers
62
views
Codistal subgroups of locally compact groups
Let $G$ be a topological group and let $H$ be a closed subgroup of $G$. Say $H$ is codistal in $G$ if the translation action of $G$ on the coset space $G/H$ is distal (meaning that no non-diagonal ...
- 4,728
2
votes
0
answers
414
views
Mixed up by definitions of mildly mixing
Here are two setup where the notion of "mildly mixing" comes up: for representations and for group acting by measure preserving transformations (see definitions below). Since a natural class of ...
- 4,432
2
votes
0
answers
103
views
Distal actions on coset spaces
Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point ...
- 4,728
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
0
answers
88
views
Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
1
vote
0
answers
107
views
Relations between $\Omega$-groups, locally indicable groups, and right-orderable groups
We know that the class of right-orderable groups $\mathit{RO}$, is contained in the class of $\Omega$-groups (read it from "A note on group rings of certain torsion-free groups" by Burns-Hale).
A ...
- 51
1
vote
0
answers
172
views
Generalizing approximate $\mathbb{Z}$-equivariance of a simple function
Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
- 15.4k
1
vote
0
answers
121
views
Name/terminology for a relationship between group actions
Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this ...
- 3,115
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101