Search Results

For Q2, the answer is yes. It suffices to enumerate the vertices of the Cayley graph by natural numbers so that each vertex (except the first one) is adjacent to at least one with a smaller number and …

It is amazing how a fact that I was taught in a middle school can be proved using big theories where I don't understand half of the words. Let me add a straightforward proof (for $S_n$ and only $S_n$) …

Concerning the second question (about 2-cycles). If some transposition (= 2-cycle) can be obtained as as a product of a polynomial number of generators (= elements of $A\cup A^{-1}$), then all element …

Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic Cayle …

I think the following is sufficiently elementary: a transposition in $S_n$ is an element of order 2 commuting with at least $2(n-2)!$ elements of the group. But $A_{n+1}$ does not have such an element …

Here is a counter-example with $G$ homeomorphic to $\mathbb R^2$. Let $f:\mathbb R\to\mathbb R$ be a discontinous additive homomorphism (constructed using a Hamel basis of $\mathbb R$ over $\mathbb Q$ …

No. Consider solid angles in $\mathbb R^2$. They all (except the half-plane) are isomorphic, yet one may be strictly included in another. Furthermore, a generic cone in $\mathbb R^d$, $d\ge 3$, has a …

Upon thinking about this question, I have a feeling that there is an interesting general problem like that, but I cannot verbalise it. Here is an approximation. The question is: given a finitely gene …

Here is a direct proof for free groups. Let $x_1,\dots,x_m$ be the generators of our group. Consider a word $x_{i_n}^{e_n}\dots x_{i_2}^{e_2}x_{i_1}^{e_1}$ where $e_i\in\{\pm 1\}$ and there are no ca …

A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth m …

Let $G=\mathbb Z^2$. Every invariant linear order on $\mathbb Z^2$ is either induced from the standard order on $\mathbb R$ (or its inverse) by a linear map of the form $$ (x,y)\mapsto x+\alpha y : \ …