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Probability and Geometry on Groups Lecture Notes for a Graduate Course

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Probability and Geometry on Groups Lecture Notes for a Graduate Course Probability and Geometry on Groups Lecture notes for a graduate course Gabor´ Pete Technical University of Budapest http://www.math.bme.hu/~gabor September 11, 2020 Work in progress. Comments are welcome. Abstract These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto. The main goal is to give a self-contained introduction to several interrelated topics of current research interests: the connections between 1) coarse geometric properties of Cayley graphs of infinite groups; 2) the algebraic properties of these groups; and 3) the behaviour of probabilistic processes (most importantly, random walks, harmonic functions, and percolation) on these Cayley graphs. I try to be as little abstract as possible, emphasizing examples rather than presenting theorems in their most general forms. I also try to provide guidance to recent research literature. In particular, there are presently over 250 exercises and many open problems that might be accessible to PhD students. It is also hoped that researchers working either in probability or in geometric group theory will find these notes useful to enter the other field. Contents Preface 4 Notation 5 1 Basic examples of random walks 6 d 1.1 Z and Td, recurrence and transience, Green’s function and spectral radius ......... 6 1.2 Large deviations: Azuma-Hoeffding and relative entropy . .............. 13 2 Free groups and presentations 18 2.1 Introduction...................................... ....... 18 2.2 Digression to topology: the fundamental group and covering spaces.............. 20 1 2.3 Themainresultsonfreegroups. ......... 23 2.4 PresentationsandCayleygraphs . .......... 25 3 The asymptotic geometry of groups 27 3.1 Quasi-isometries. Ends. The fundamental observation of geometricgrouptheory . 27 3.2 Gromov-hyperbolicspacesandgroups . ........... 32 3.3 Asymptoticcones................................... ....... 32 4 Nilpotent and solvable groups 32 4.1 Thebasics......................................... ..... 33 4.2 Semidirectproducts ................................. ....... 35 4.3 The volume growth of nilpotent and solvable groups . ........... 39 4.4 Expanding maps. Polynomial and intermediate volume growth . ............ 42 5 Isoperimetric inequalities 44 5.1 Basic definitions and examples . ....... 44 5.2 Amenability, invariant means, wobbling paradoxical decompositions ............. 48 5.3 Fromgrowthtoisoperimetryingroups. .......... 50 5.4 Isoperimetry in Zd and 0, 1 n .................................. 51 { } 6 Random walks, discrete potential theory, martingales 53 6.1 Markov chains, electric networks and the discrete Laplacian . ............... 53 6.2 Dirichlet energy and transience . ......... 59 6.3 Martingales ........................................ ..... 62 7 Cheeger constant and spectral gap 68 7.1 SpectralradiusandtheMarkovoperatornorm . ............. 68 7.2 The infinite case: the Kesten-Cheeger-Dodziuk-Mohar theorem ................ 71 7.3 Thefinitecase:expandersandmixing . ......... 73 7.4 From infinite to finite: Kazhdan groups and expanders . ............. 84 8 Isoperimetric inequalities and return probabilities in general 86 8.1 Poincar´eand Nash inequalities . ........ 87 8.2 Evolvingsets:theresults ............................. ........ 88 8.3 Evolvingsets:theproof .............................. ........ 90 9 Speed, entropy, Liouville property, Poisson boundary 95 9.1 Speedofrandomwalks................................ ....... 95 9.2 The Liouville and strong Liouville properties for harmonic functions ............. 99 9.3 Entropy,andthemainequivalencetheorem . ........... 103 9.4 Liouville and Poisson . ..... 106 9.5 The Poisson boundary and entropy. The importance of group-invariance. 110 9.6 Unboundedmeasures................................ ........ 113 2 10 Growth of groups, of harmonic functions and of random walks 116 10.1 AproofofGromov’stheorem . ......... 116 10.2 Random walks on groups are at least diffusive . ........... 120 11 Harmonic Dirichlet functions and Uniform Spanning Forests 121 11.1 Harmonic Dirichlet functions . ........ 121 11.2 Loop-erased random walk, uniform spanning trees and forests................. 122 12 Percolation theory 126 12.1 Basic definitions, examples and tools . .......... 126 12.2 Percolation on infinite groups: pc, pu, unimodularity, and general invariant percolations . 139 12.3 Percolation on finite graphs. Threshold phenomema . ............. 154 12.4 Critical percolation: the plane, trees, scaling limits, critical exponents, mean field theory . 163 12.5 Geometry and random walks on percolation clusters . ............. 176 13 Further spatial models 181 13.1 Ising,Potts,andthe FK randomcluster models . ............. 181 13.2 Extremal Gibbs measures and factor of IID processes . ................. 193 13.3 Bootstrap percolation and zero temperature Glauber dynamics ................ 195 13.4 MinimalSpanningForests . ........ 196 13.5 Measurable group theory and orbit equivalence . ............. 198 14 Local approximations to Cayley graphs 203 14.1 Unimodularity and soficity . ........ 203 14.2 Spectral measures and other probabilistic questions . ................ 213 15 Some more exotic groups 225 15.1 Self-similar groups of finite automata . ........... 226 15.2 Constructing monsters using hyperbolicity . ............. 232 15.3 Thompson’s group F ........................................ 232 16 Quasi-isometric rigidity and embeddings 233 References 234 3 Preface These notes have grown (and are still growing) out of two graduate courses I gave at the University of Toronto: Probability and Geometry on Groups in the Fall of 2009, and Percolation in the plane, on Zd, and beyond in the Spring of 2011. I am still adding material and polishing the existing parts, so at the end I expect it to be enough for two semesters, or even more. Large portions of the first drafts were written up by the nine students who took the first course for credit: Eric Hart, Siyu Liu, Kostya Matveev, Jim McGarva, Ben Rifkind, Andrew Stewart, Kyle Thompson, Llu´ısVena, and Jeremy Voltz — I am very grateful to them. That first course was completely introductory: some students had not really seen probability before this, and only few had seen geometric group theory. Here is the course description: Probability is one of the fastest developing areas of mathematics today, finding new connections to other branches constantly. One example is the rich interplay between large-scale geometric properties of a space and the behaviour of stochastic processes (like random walks and percolation) on the space. The obvious best source of discrete metric spaces are the Cayley graphs of finitely generated groups, especially that their large-scale geometric (and hence, probabilistic) properties reflect the algebraic properties. A famous example is the construction of expander graphs using group representations, another one is Gromov’s theorem on the equivalence between a group being almost nilpotent and the polynomial volume growth of its Cayley graphs. The course will contain a large variety of interrelated topics in this area, with an emphasis on open problems. What I had originally planned to cover turned out to be ridiculously much, so a lot had to be dropped, which is also visible in the present state of these notes. The main topics that are still missing are Gromov- hyperbolic groups and their applications to the construction of interesting groups, metric embeddings of groups in Hilbert spaces, more on the construction and applications of expander graphs, more on critical spatial processes in the plane and their scaling limits, and a more thorough study of Uniform Spanning Forests and ℓ2-Betti numbers — I am planning to improve the notes regarding these issues soon. Besides research papers I like, my primary sources were [DrKa09], [dlHar00] for geometric group theory and [LyPer16], [Per04], [Woe00] for probability. I did not use more of [HooLW06], [Lub94], [Wil09] only because of the time constraints. There are proofs or even sections that follow rather closely one of these books, but there are always differences in the details, and the devil might be in those. Also, since I was a graduate student of Yuval Peres not too long ago, several parts of these notes are strongly influenced by his lectures. In particular, Chapter 9 contains paragraphs that are almost just copied from some unpublished notes of his that I was once editing. There is one more recent book, [Gri10], whose first few chapters have considerable overlap with the more introductory parts of these notes, although I did not look at that book before having finished most of these notes. Anyway, the group theoretical point of view is missing from that book entirely. With all these books available, what is the point in writing these notes? An obvious reason is that it is rather uncomfortable for the students to go to several different books and start reading them somewhere from their middle. Moreover, these books are usually for a bit more specialized audience, so either nilpotent groups or martingales are not explained carefully. So, I wanted to add my favourite explanations and examples to everything, and include proofs I have not seen elsewhere in the literature. And there was a very important goal I had: presenting the material in constant conversation between the probabilistic and 4 geometric group theoretical ideas. I hope this will help not only students, but also researchers from either field get interested and enter the other territory. There are presently over 200 exercises,
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