In the previous week I read:

• Chapter 4.5 of Thurston’s famous book
• Section 5.1.2 of DGO’s paper of relative hyperbolic groups

This is merely a draft and I shall add something more soon (hopefully before the next weekend).

Thick-thin decomposition

The main theorem here is that, given a hyperbolic manifold, it is compact if and only if all of its thick parts are.

I believe I misrememberd some facts here. Maybe not compact but finite-volume, since I remember cusps got a say also.

Windmill

… and Don Quixote.

Well, imagining I have a bunch of rotation acting on a space and they are fay away from each other. That is to say, the pivot, or the point that the fixed, are far way from each other. If I look at the group action on this space, I’d expect it to be the free-product of all those rotation groups, right? Because it certainly contains all of them and their combinations. Also, since they are fay away from each other, I’d expect their action do not interfere with each other.

That’s gthe big theorem here. The windmill is constructed by starting with one pivot and its neighborhood, and extend it a little bit, look at its orbit under the rotation groups, and keep doing this. The astonishing thing here is that it (potentially intentially) mimics the construction of Bass-Serre group, which claims that the result would be the free product of all those rotation groups.

Given a space, we want to study it “step by step”. We want to sort out the points, the edges, the faces etc. All those things are connected: edges connecting points, faces gluing to edges etc, and those connecting can be viewed as maps. In algebra, such a long chain is called a chain complex, and our first goal is to assign a chain complex to a given space. Then we can use homology to understand the maps.