# Tagged Questions

**1**

vote

**0**answers

100 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 ...

**25**

votes

**2**answers

1k views

### Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group
generated by the permutation
$$
a: \ (m,n) \ \mapsto \ (m-n,m)
$$
of order $6$ and the involutions
$$
b: \ (m,n) \ ...

**1**

vote

**0**answers

171 views

### Examples of amenable groups acting on the real line

In the literature, there are several examples of solvable groups acting by order preserving homeomorphisms of the real line. There are also examples of groups of intermediate growth acting in the same ...

**6**

votes

**2**answers

317 views

### Rotation numbers for amenable group actions on the circle

Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + ...

**11**

votes

**0**answers

148 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 ...

**3**

votes

**0**answers

84 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 ...

**5**

votes

**2**answers

234 views

### pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...

**8**

votes

**2**answers

364 views

### Birational Automorphisms and infinite divisibility

Suppose $X$ is some algebraic variety. It can be over $\mathbb{C}$, but it doesn't have to (but char $0$ preferred).
Is it possible that the additive group $\mathbb{Q}$ acts on it birationally, ...

**3**

votes

**0**answers

243 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,
...

**4**

votes

**2**answers

269 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 ...

**3**

votes

**2**answers

422 views

### Another question about amenability and Følner sequences

Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$,
...

**6**

votes

**4**answers

601 views

### Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$

The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace ...

**9**

votes

**0**answers

262 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 ...

**2**

votes

**0**answers

107 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 ...

**1**

vote

**4**answers

316 views

### A Fractional Linear Transformation Class Property

Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and ...

**4**

votes

**0**answers

112 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 ...

**1**

vote

**2**answers

949 views

### Periodic matrices in SL(3,Z)

Periodic matrices in SL(3,Z) will be conjugated to
product of periodic matrices in SL(2,Z) by +- indentity on a third
integer direction. Is this true?
Sorry, following your comments, maybe ...

**7**

votes

**1**answer

421 views

### Classification of countable subgroups of the circle

Is there a classification of all countable subgroups of the circle $\mathbb{T} \simeq \mathbb{R}/\mathbb{Z}$?
It seems that there are quite a lot of them, e.g.:
cyclic subgroups $\{a^n\colon ...

**14**

votes

**1**answer

864 views

### Rokhlin lemma for arbitrary infinite groups.

Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action ...

**6**

votes

**1**answer

532 views

### Actions orbit equivalent to profinite ones

Let $G$ be a countable discrete residually finite group.
Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?
Ozawa and Popa introduced the concept of ...

**6**

votes

**0**answers

296 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 ...

**6**

votes

**3**answers

1k views

### amenable equivalence relation generated by an action of a non-amenable group

Question. Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ ...

**16**

votes

**13**answers

3k views

### Is there a “crash-course” book on Abelian varieties (e.g., an introduction for physicists)?

Hello,
In our (rather applied) theoretical physics research, we have encountered an important class of problems, which seem to require an understanding of Abelian functions (unfortunately, this ...

**2**

votes

**1**answer

687 views

### Are all Nilmanifolds quotients of Heisenberg Group

I've been reading some wonderful blog entries where Terry Tao and Ben Green prove some generalizations of Weyl Equidstribution using a "higher" Fourier Analysis. Unfortunately, all the information I ...

**26**

votes

**3**answers

2k views

### Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...

**6**

votes

**2**answers

272 views

### Homomorphisms from R to $Diffeo^+(R)$, or “fractional iterations”

Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a ...

**5**

votes

**1**answer

343 views

### Automorphisms of $\pi_1$ induced by pseudo-Anosov maps

Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.
For any ...