An interesting thing about these combinatorial maps is that there are "sequences" of permutations which sort of draw these maps, and studying these was crucial in the proof of the "four color theorem" for higher genus surfaces
this Mj guy thinks it's a good question. i can see why it is so : having such a simply connected space would give me a space X/Gal(Q) which has fundamental group Gal(Q) and all higher "homotopy" groups trivial, so that'd be a K(Gal(Q), 1)-space, and (co)homology of this space would return Galois (co)homology, which'd be nice.
don't take what i say literally, since usual notion of fundamental groups won't work.
anyway, these are all very hard questions. better to study some more stuff before thinking about them.
@DiscipleofBarney yeah, i mean, is there a sensible way to define hyperbolicity for profinite groups? you can't draw Cayley graphs since these are general uncountable.
the correct way to do it probably is once you have a group $G = \varprojlim G_i$ then draw the Cayley graphs $\Gamma(G_i)$ of each group in the set and then take inverse limit of the graphs.
for example if you have $\mathbf{Z}_p = \varprojlim \Bbb Z/p^n\Bbb Z$ then the corresponding "graph" would be inverse limit of a bunch of cycles. doing it the "right" way, one ends up with the $p$-adic solenoid. but there are also some boring way, where you end up with just the real line :p
I am looking through it (using some search functionality) and so far it seems they are referencing a paper, I am still looking though.
An LC-group G is Gromov-hyperbolic if it is compactly generated and if, with a word metric, it is a Gromov-hyperbolic metric space; equivalently if G has a continuous proper cocompact isometric action on some proper geodesic hyperbolic metric space [CCMT, Corollary 2.6].
Page 103 in that book (at least when I grabbed it, it may have been expanded since)
that's too bad. i don't think $\mathbf{Z}_p$ acts on any proper hyperbolic geodesic metric space. (Hilbert-Smith conjecture at least says that it can't act on a manifold)
Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas Monod, and Romain Tessera, Amenable hyperbolic groups, J. Eur. Math. Soc., to appear. 49, 51, 56, 103, 108
Remark 2.5. The assumption that L is a Lie group is essential in Lemma 2.4. Indeed, let G = R × Zp, where Zp denotes the (compact) additive group of the p-adic integers. Let Z be a copy of Z embedded diagonally in G, and let L = G/Z be the quotient group. The group L is the so-called solenoid and can alternatively be defined as the inverse limit of the iterated p-fold covers of the circle group. It is connected (but not locally arcwise connected). The image of G◦ = R under the quotient map π : G → L is dense, but properly contained, in L.
"To coursework master's, honours, or keen third year students: If you're looking for a maths course to take next semester, I'm currently putting out feelers for people interested in algebraic topology. Joe Grotowski has offered to run a reading course (how you credit this course would need to be worked out by you with the faculty, although it is normally taken as 'special topics'). He said he will reluctantly run it for two people, but would be happy to run it for three or more people, which means that I'd like at least two more people."
@Incurrence I don't know much topology, but I think hatcher is considered one of the more accessible treatments with lots of motivation, pictures, and geometry. In fact I know you had plans to go through A concise course, but the author of that book recommends it as something to go alongside another book, like hatcher
I have not worked through hatcher, although in a week or two I will be starting the homology chapter of that book
@ᴇʏᴇs If it says to prove something is defined on all of $\Bbb C$ but nowhere analytic, they just mean that it fails the cauchy riemann equations at all points, but has no points at infinity right?
@Incurrence Yah, trying to get enough of this small cancellation theory stuff so that I can finally solve that damn problem I was talking about with BalarkaSen. If I manage to figure it out, I will have a tweaking on math for days and won't be able to sleep. Maybe Tweaking Tank will be my new name :D
One of my current students wants to "self-learn" abstract algebra so that he can take cryptography in the fall. I've strongly discouraged that. Many students are unrealistic and overestimate their own capabilities.
well, mr eyeglasses, I sure wish some of my students would do some self-learning of the course I'm teaching; they're certainly not making any effort to learn it from me.
