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Random walks on weakly hyperbolic groups

Random and Arithmetic Structures in Topology: Introductory Workshop August 25, 2020 - September 11, 2020

September 01, 2020 (10:30 AM PDT - 11:30 AM PDT)
Speaker(s): Giulio Tiozzo (University of Toronto)
Location: MSRI: Online/Virtual
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

Random Walks On Weakly Hyperbolic Groups 2


The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). 

Starting in the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices. 

In this course, we will consider random walks on groups of isometries of delta-hyperbolic spaces,  and establish their asymptotic properties: for instance, sample paths almost surely converge to the boundary and have positive drift. 

In recent years, this has had many applications to low-dimensional topology,  as e.g. the mapping class group and Out(F_n) act on certain (non-locally compact) hyperbolic spaces. We will discuss some such applications and their relations to Teichmuller theory. 

Lecture plan: 


1) Introduction to delta-hyperbolic spaces and random walks

2) The horofunction boundary 

3) Convergence to the boundary and positive drift

4) Genericity of hyperbolic elements


The course is mostly based on my joint work with J. Maher. 

Asset no preview Notes (Part 2) 406 KB application/pdf Download
Asset no preview Lecture 2 (Slides) 560 KB application/pdf Download
Asset no preview Lecture 1 (Slides) 751 KB application/pdf Download
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Random Walks On Weakly Hyperbolic Groups 2

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