Category Archives: Brief Introduction (BI)

BI: the Bestvina-Bromberg-Fujiwara construction

This post is about a remarkable construction due to Bestvina-Bromberg-Fujiwara. It has been first used to show that the asymptotic dimension of Mapping Class Groups is finite, and it is quite useful, for example, in this great paper by Dahmani-Guirardel-Osin … Continue reading

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Hyperbolicity of the curve graph: the proof from The Book

In this post I’ll talk about a lovely paper by Sebastian Hensel, Piotr Przytycki and Richard Webb. They show that all curve graphs are 17-hyperbolic. Hyperbolicity of curve graphs is a very very very useful* property because mapping class groups … Continue reading

Posted in guessing geodesics, hyperbolicity of complexes | 3 Comments

BI: Finite decomposition complexity (is preserved by relative hyperbolicity)

In this post I’ll define finite decomposition complexity (FDC) for you, tell you what it’s good for (Stable Borel Conjecture!) and point out that an argument by Osin shows that it is preserved by relative hyperbolicity. First, the motivation. Here … Continue reading

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BI: Guessing geodesics in hyperbolic spaces

In this post I’ll discuss a cool lemma due to Brian Bowditch (from this paper) which is very useful when you want to show that a space is hyperbolic. The most direct way of showing that is hyperbolic is trying … Continue reading

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BI: Teichmüller space, part I

It is a remarkable fact that the group of orientation-preserving isometries of can be identified with the group of bi-holomorphisms of the unit disk in . [Fun fact: Poincaré said that he discovered (basically) this while stepping on a bus.] … Continue reading

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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

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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

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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

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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

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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

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