User Paul Plummer | chat.stackexchange.com

Paul Plummer

Sep 16, 2020 21:57

There is definitely more to show with Balarka idea but I suspect if you dissected everything you could probably translate it to that

Paul Plummer

Sep 16, 2020 21:55

I don't know @MikeMiller

Paul Plummer

Sep 16, 2020 21:53

From what I understand all the ball model stuff that we know goes through strongly contracting property stuff and doesn't appear to apply when we only have weakly contracting (it is open if there are strongly contracting geodesics)

Paul Plummer

Sep 16, 2020 21:51

Yah, I think the idea is basically the same

Paul Plummer

Sep 16, 2020 21:50

There is also other "intrinsic" hyperbolic behavior. For example you can get a quasi-axis orbit from pA acting on the mapping class group and these axis have a property called (weakly) contracting, which basically means balls disjoint from the axis project to "small" diameter sets. For hyperbolic spaces there is a uniform bound and when that happens you get the strongly contracting property

Paul Plummer

Sep 16, 2020 21:48

Hi

Paul Plummer

Sep 16, 2020 21:47

pA's are also exactly the hyperbolic elements acting on the curve graph, which has boundary as the space of ending laminations

Paul Plummer

Sep 16, 2020 21:46

and pA have north south dynamics.

Paul Plummer

Sep 16, 2020 21:45

There isn't a right boundary for MCG but there are useful ones. Teichmuller space has projective measured laminations as a boundary

Paul Plummer

Sep 16, 2020 21:44

(and I don't know the proof, I know it uses teichmuller theory stuff)

Paul Plummer

Sep 16, 2020 21:43

Showing that all finite subgroups arise as the isometry group of some surface is harder

Paul Plummer

Sep 16, 2020 21:42

but it is supposed to all be analogous to the torus case where teichmuller space is the hyperbolic plane

Paul Plummer

Sep 16, 2020 21:42

I think it takes about a page in the primer (Farb and Margalit) once you have all the definitions and stuff

Paul Plummer

Sep 16, 2020 21:40

@MikeMiller It isn't exactly easy to see and does take work. The classification basically goes through the action on Teichmuller space, finite order case corresponds exactly to fixing a point

Paul Plummer

Sep 16, 2020 20:36

@BalarkaSen Here is an interesting open problem. In MCG of surfaces there is a classification into elements finite order, reducible, irreducible/pseudo-Anosov, and the irreducible elements have lots of "hyperbolic behavior". It is and open question whether or not these are generic in the ball model. Weirdly it is known that a positive proportion are irreducible in the ball model.

It has been known for a while that random walks generically end at irreducible.

Paul Plummer

Sep 16, 2020 20:30

It is pretty elementary, just uses a bit about measurable sets

Paul Plummer

Sep 16, 2020 20:28

For example, showing that hyperbolic elements are generic by counting inside radius $n$ balls

Paul Plummer

Sep 16, 2020 20:27

I was surprised that the "ball model" of randomness is so hard to prove stuff in @BalarkaSen

Paul Plummer

Sep 16, 2020 20:26

@AlessandroCodenotti The "unqualified" question where it is proved that there are weird partitions (using choice) and this question

Paul Plummer

Sep 16, 2020 20:24

Ah, random walk stuff, know very little of that

Paul Plummer

Sep 16, 2020 20:24

It is all just intuition anyways :D

Paul Plummer

Sep 16, 2020 20:23

I have gotten by with very little

Paul Plummer

Sep 16, 2020 20:23

Coarse hyperbolic geometry?

Paul Plummer

Sep 16, 2020 20:21

@AlessandroCodenotti I recently answered a question where someone was wondering about partitioning reals into borel subsemigroups. Turns out there are not really interesting partitions when you restrict to such subsets.

Paul Plummer

Sep 16, 2020 20:18

Has he said anything about C-T maps not existing for some subgroups?

Paul Plummer

Sep 16, 2020 20:17

Makes sense, he is like the leader in C-T maps

Paul Plummer

Sep 16, 2020 20:15

or like a lecture

Paul Plummer

Sep 16, 2020 20:15

@BalarkaSen Are you reading a paper on it?

Paul Plummer

Sep 16, 2020 20:15

That is cool stuff

Paul Plummer

Sep 16, 2020 20:11

You are done with your thesis right @AlessandroCodenotti?

Paul Plummer

Sep 16, 2020 20:10

It is sort of GGT mixed with low dim geometry/topology

Paul Plummer

Sep 16, 2020 20:09

Hi

Paul Plummer

Sep 16, 2020 20:09

And the known proof actually uses a similar statement already proved for MCG, which is a bit weird since normally proofs go in the other direction (understand geometry of curve graph can give understanding of MCG)

Paul Plummer

Sep 16, 2020 20:08

This is already known, but I think there is a more elementary proof

Paul Plummer

Sep 16, 2020 20:07

Most the the GGT stuff I work on is in someway related, or maybe only inspired by, work on mapping class groups of surfaces. One thing is I am trying to come up with a new proof that quasiisometry of a curve graph of a surface (V=isotopy classes of curves, edge if they have disjoint representatives) is finite distance from an actual graph automorphism

Paul Plummer

Sep 16, 2020 20:01

I should have learned the classical case by now... not sure why I haven't

Paul Plummer

Sep 16, 2020 19:59

Cool, I don't think I know the proof, is there a nice idea to it? I know that normal subgroups are distorted so it isn't totally obvious

Paul Plummer

Sep 16, 2020 19:57

Yah @AlessandroCodenotti

Paul Plummer

Sep 16, 2020 19:56

A shorter period of time than the last gap at least

Paul Plummer

Sep 16, 2020 19:56

Hey

Paul Plummer

Aug 3, 2020 03:40

That example points out that even though we have morse lemma doesn't mean it is "syncronous"

Paul Plummer

Aug 3, 2020 03:36

Also for that quasi-isometry finite dist from automorphism here is a silly example: consider the integers and the quasi isometry is multiply by 2. it isn't finite dist because of d(2,4)=2,d(4,8)=4,... I will think a bit for a more interesting example

Paul Plummer

Aug 3, 2020 03:22

@BalarkaSen And a blog post sketching the already short proof: alexsisto.wordpress.com/2013/02/08/…

Paul Plummer

Aug 3, 2020 03:22

Oh also there is a modern combinatorial proof of hyperbolicity: arxiv.org/abs/1301.5577

Paul Plummer

Aug 3, 2020 02:56

The above is a note by Minsky before he published the papers about all this with Masur

Paul Plummer

Aug 3, 2020 02:55

citeseerx.ist.psu.edu/viewdoc/…

Paul Plummer

Aug 3, 2020 02:53

Also in teichmuller space the axis for pseudo Anosov elements have a morse lemma, so it has "hyperbolic directions"

Paul Plummer

Aug 3, 2020 02:53

A remarkable statement is that in some sense you can coarsely get the distance in MCG by adding up distances in all the curve graphs (sort of, needs more explanation to even make sense)

Paul Plummer

Aug 3, 2020 02:50

Now MCG is not relatively hyperbolic but this action is nice enough you do get information, and this "inductive" procedure can help give even more info

Paul Plummer

Aug 3, 2020 02:50

So part of the idea is well if I collapse all this stuff in some meaningful way you get some space, which is the curve graph

Paul Plummer

Mathematics

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