Acknowledgments xiii Given a subset E of a space X, the indicator function 1E : X → R is defined by setting 1E(x) equal to 1 for x ∈ E and equal to 0 for x ∈ E. The cardinality of a finite set E will be denoted |E|. We will use the asymptotic notation X = O(Y ), X Y , or Y X to denote the estimate |X| ≤ CY for some absolute constant C 0. In some cases we will need this constant C to depend on a parameter (e.g., d), in which case we shall indicate this dependence by subscripts, e.g., X = Od(Y ) or X d Y . We also sometimes use X ∼ Y as a synonym for X Y X. (Note though that once we deploy the machinery of nonstandard analysis in Chapter 7, we will use a closely related, but slightly different, asymptotic notation.) Acknowledgments I am greatly indebted to my students of the course on which this text was based, as well as many further commenters on my blog, including Marius Buliga, Tony Carbery, Nick Cook, Alin Galatan, Pierre de la Harpe, Ben Hayes, Richard Hevener, Vitali Kapovitch, E. Mehmet Kiral, Allen Knutson, Mateusz Kwasnicki, Fred Lunnon, Peter McNamara, William Meyerson, Joel Moreira, John Pardon, Ravi Raghunathan, David Roberts, David Ross, Olof Sisask, David Speyer, Benjamin Steinberg, Neil Strickland, Lou van den Dries, Joshua Zelinsky, Pavel Zorin, and several anonymous commenters. These comments can be viewed online at: terrytao.wordpress.com/category/teaching/254a-hilberts-fifth-problem/ The author was supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award. Last, but not least, I thank Emmanuel Breuillard and Ben Green for introducing me to the beautiful interplay between geometric group theory, additive combi- natorics, and topological group theory that arises in this text.

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