Category Archives: Brief Introduction (BI)

BI: the Bestvina-Bromberg-Fujiwara construction

Posted on December 11, 2013 by

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

Posted on February 8, 2013 by

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)

Posted on October 3, 2012 by

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

Posted on July 16, 2012 by

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 →

Posted in guessing geodesics | 2 Comments

BI: Teichmüller space, part I

Posted on May 6, 2012 by

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

Posted on February 19, 2012 by

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

Posted on February 5, 2012 by

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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