Author Archives: Alex Sisto

EO: embedding graph manifold groups in products of trees

This post is based on this paper with David Hume. As discussed here, it is a nice property for a group to be quasi-isometrically embeddable in a product of finitely many trees. Graph manifolds are certain 3-manifolds, and if you … Continue reading

Posted in graph manifolds in products of trees | Leave a comment

EO: contracting elements

In this post I’ll talk about this paper. There are many groups that have been known for quite a while to “look like” hyperbolic groups, but only in certain “directions”. Examples of such directions are (the axes of ) rank … Continue reading

Posted in contracting elements | Leave a comment

Why not donating your paper to the public domain?

Breaking news: David Hume and I just managed to make sure that our paper will be in the public domain when published (by Proceedings of the AMS). We have been inspired by Saul Schleimer, who has been consistently dedicating his papers … Continue reading

Posted in Why not donating your paper to the public domain? | Leave a comment

BI: nonstandard analysis, a small investment

First of all, you can read about nonstandard analysis in a slightly more famous blog than my own, for example you can check out this post. I guess I should explain the quote “a small investment” by Isaac Goldbring (afaik). … Continue reading

Posted in nonstandard analysis | Leave a comment

BI: Embedding hyperbolic spaces in products of trees

Lately, I’m having fun trying to quasi-isometrically embed metric spaces of all sorts in products of trees, like in this paper with David Hume (!). [EDIT: and this paper with John MacKay] This hobby started when David told me about this … Continue reading

Posted in embeddings in products of trees | Leave a comment

BI(+EO): quasi-isometric rigidity

There are a few types of quasi-isometric rigidity results. I will talk about results of the following kinds: 1) self-quasi-isometries of such-and-such space or group (coarsely) preserve some structure, usually a collection of (left cosets) of subgroups 2) if a … Continue reading

Posted in higher-dimensional graph manifolds, quasi-isometric rigidity | Leave a comment

BI: Asymptotic cones

This post contains an experimental exposition of asymptotic cones, I hope this will be vaguely understandable (feedback is warmly welcome!). Many (most?) researchers actually think of asymptotic cones in a way similar to the one I’ll present, rather than as … Continue reading

Posted in asymptotic cones | Leave a comment

EO: “hierarchy paths” for graph manifold groups

I’d like to tell you about a certain family of bilipschitz paths in universal covers of graph manifolds with nice “combinatorial” properties that I used here and later here, and that hopefully will turn out to be useful in other … Continue reading

Posted in special paths in graph manifolds | 1 Comment

BI: Thickness

Thick metric spaces were defined in this paper by Jason Behrstock, Cornelia Druţu and Lee Mosher as an obstruction for a group (actually, a metric space) to be (properly) relatively hyerbolic. Indeed, the notion of thickness provides a very interesting way of … Continue reading

Posted in Brief Introduction (BI), thickness | Tagged | 1 Comment

EO: quasi-isometric embeddings of the hyperbolic plane in relatively hyperbolic groups

This post is based on a joint work with John MacKay. It is still an open question whether or not all non-virtually-free hyperbolic groups contain a surface group. This is actually related to (and AFAIK inspired by) an important open … Continue reading

Posted in Expanded Outline (EO), quasi-hyperbolic-planes | Tagged | Comments Off on EO: quasi-isometric embeddings of the hyperbolic plane in relatively hyperbolic groups