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Author Archives: Alex Sisto
What is a hierarchically hyperbolic space?
Hey everybody, it’s been a while… This post is about the notion of hierarchically hyperbolic space we defined with Jason Behrstock and Mark Hagen in this paper, which is a generalisation of the notion of (Gromov)hyperbolic space. The main examples … Continue reading
Posted in HHS
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BI: the BestvinaBrombergFujiwara construction
This post is about a remarkable construction due to BestvinaBrombergFujiwara. 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 DahmaniGuirardelOsin … Continue reading
An even shorter proof that curve graphs are hyperbolic
In this post I described the HenselPrzytyckiWebb proof that curve graphs are (uniformly) hyperbolic. Their methods actually apply to the arc graph, and then they used a clever evil trick due to Harer that involves adding an artificial puncture to … Continue reading
Posted in hyperbolicity of complexes
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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 17hyperbolic. Hyperbolicity of curve graphs is a very very very useful* property because mapping class groups … Continue reading
Tracking of random walks with geodesics
In this post I’ll tell you about a property of random walks on hyperbolic groups. To make a random walk on a group just start from the identity in the Cayley graph, then move to a neighbor with uniform probability … Continue reading
Posted in random walks
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Just for fun: genus 2 madness in Bonn
Real life model of the Lsurface, realised by Mark Pedron with the help of Dawid Kielak. Yeah, too much light, I know… Clearer one: Explanation on the blackboard: The cone point. Behold the negative curvature!
Posted in Just for fun
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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