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9 votes
Accepted

Commutation classes of reduced decompositions of the longest element of the Weyl group with one element

The reduced words that are in their own commutation classes are: $s = [123\cdots(n-2)(n-1)(n-2)\cdots321][23\cdots(n-3)(n-2)(n-3)\cdots32][3\cdots(n-4)(n-3)(n-4)\cdots3]\cdots$ the reversal of $s$ (...
7 votes

Commutation classes of reduced decompositions of the longest element of the Weyl group with one element

I believe that for $n \geq 4$ there will be exactly $4$ such reduced words. One such word, call it $R_n$ can be constructed by starting with $s_{n-1}s_{n-2} \cdots s_2s_1s_2 \cdots s_{n-2}s_{n-1}$ and ...
1 vote

Group actions and "transfinite dynamics"

Here is a partial answer extracted from a Twitter user's answer that works for actions with uniformly bounded finite orbits, which indeed solves a special case that initially motivated this question. ...
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2 votes

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

The centre of any non-cyclic one-relator group with torsion is trivial. Moreover, the centraliser of any non-identity element of a one-relator group with torsion is cyclic. This is B. B. Newman’s ...
7 votes

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

The groups have trivial center, as pointed out by Sam Nead. Another, more combinatorial, way to show this is to apply the algorithm from [1], which decides whether any given one-relator group has a ...
8 votes
Accepted

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

These groups have trivial centers. As one proof, they are both fuchsian and so embed in $\mathrm{PSL}(2, \mathbb{R})$ (well, the orientation preserving subgroups do). However, elements of $\mathrm{...
  • 21k
8 votes

Is there a flat manifold with trivial first homology?

Andrzej Szczepański pointed me to Proposition 2.3.13 in the book [Perfect Groups, Derek F. Holt and Wilhelm Plesken, 1989], which gives an answer to my question. Namely, in a slightly different ...
1 vote

Proving that a countable group is not finitely generated

One possible way of proving that a countable group $G$ is not finitely generated is finding an infinite set $S$ and a mapping $\varphi$ from $G$ to the power set of $S$ such that the following hold: $...
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0 votes
Accepted

Finitely generated groups with Hölder-exotic space of ends?

The question is solved positively in the paper The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups by Matthieu Dussaule, Wenyuan Yang, which appeared on arXiv on October ...
1 vote

Groups with three conjugacy classes that define an ordering

I have proposed a positive solution to this problem in a preprint entitled Hyperexponentially closed fields, to be found here, more precisely in Sections 10.1 and 10.2. The solution is based on work ...
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5 votes
Accepted

Injectivity of the cohomology map induced by some projection map

Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture. The ...
  • 8,728
4 votes

Injectivity of the cohomology map induced by some projection map

OK, just noticed mme's comment, which considers the very same counterexample below with a clean one-line argument. I'm leaving this here in case someone finds something of any value, but I think mme's ...
3 votes
Accepted

Catalogue of groups with short finite presentations

I would very much like to have such a database and would like to contribute to its development. Prompted by this question, we talked about what such a database could look like (e.g. in terms of groups ...
  • 1,956
2 votes
Accepted

Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts

I don't know that much literature on the multidimensional case (though I'm not sure I'm the one who would if there is literature, either), but I can collect the comments and try to add a few things. ...
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0 votes
Accepted

Kronecker product preserves the conjugacy relation?

If $A$ and $B$ are elementary abelian $2$-subgroups of $\mathrm{PGL}_n(\mathbf C)$ of rank $r$ then they lift uniquely to elementary abelian $2$-subgroups of $\mathrm{GL}_n(\mathbf C)$ of rank $r+1$ (...
1 vote

Axioms for the category of groups

The category of groups is the universal example of a cocomplete category equipped with a cogroup object. A similar statement holds for other types of algebraic structures. This is due to Freyd. See ...
2 votes
Accepted

Density of “diagonal sets” in amenable groups

The answer to your question as stated is "no", but a variant of it is true (see the proposition below). Proof that the answer is "no": Let $(F_n)$ be the Følner sequence in $\...
5 votes
Accepted

Question about maximal compact subgroups of Lie groups

$\DeclareMathOperator\Lie{Lie}\newcommand\g{\mathfrak g}\newcommand\C{{\mathbb C}}$$\g = \Lie(G)$ is maximal among subalgebras of $\g_\C = \Lie(G_\C)$ on which the Killing form is negative definite, ...
  • 9,057
6 votes

Does a perfect $4^{11}\cdot M_{24}$ exist?

I think the answer is no. In fact there are no perfect split extensions with structure $4^{11}:M_{24}$. I deduced this from some cohomology calculations in Magma. Note that there are two (mutually ...
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2 votes
Accepted

Representation of $\mathrm{AGL}(V)$ on the homology of the poset of affine subspaces of $V$

I suspect that Solomon, Louis The affine group. I. Bruhat decomposition. proves what you are looking for. Let $A_n(q)$ denote the poset of proper affine subspaces of $\mathbf{F}_q^n$. The only non-...
4 votes

Bandwidth of finite groups

There is a well-known conjecture of Babai: suppose that $G$ is a finite, non-abelian, simple group. Suppose that $S$ is any generating set for $G$. Then the diameter of the resulting Cayley graph is ...
  • 21k
4 votes
Accepted

Amenable subsets of groups

Here is a slightly modified version of an example that was communicated to the author of the question by Nicolas Monod. Let $k>1$ be an integer and $G$ the Baumslag-Solitar group $\mathrm{BS}(1,k)$;...
4 votes

Fusing conjugacy classes II

No. Take $G={\rm SL}(2,{\Bbb R})$, $\ H=\{\,h(\lambda)={\rm diag}(\lambda, \lambda^{-1})\ |\ \lambda\in {\Bbb R}, \lambda>0\,\}$, $$U=\bigg\{ u(a)= \begin{pmatrix} 1 &a\\ 0&1 \end{pmatrix}\...

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