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.