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Discrete Groups, Expanding Graphs and Invariant Measures

Publisher: 
Birkhäuser
Number of Pages: 
191
Price: 
39.95
ISBN: 
9783034603317

The redoubtable Freeman J. Dyson started his marvelous career as a number theorist, having studied with G. H. Hardy in the dark days of World War II. Post bellum he underwent a metamorphosis into a quantum field theorist and famously became the conduit between Julian Schwinger and Richard Feynman; there are some who believe that he should have been awarded a share of the 1965 Nobel Prize with Schwinger, Feynman, and Tomonaga. After coming to the States, Dyson was a physicist, and a number theorist no longer, his early work at Cambridge notwithstanding.

However, the more things change the more they stay the same: in 1972 Dyson met the youthful Hugh Montgomery at a Fuld Hall social at the Institute for Advanced Study, and, upon learning of the latter’s recent work on the zeta function, contributed the now famous insight that led to the exciting formulation of the Riemann hypothesis in terms of eigenvalues of random matrices, an area now associated also with Peter Sarnak, who will appear on the stage soon.

So Dyson, the great theoretical physicist, is truly also much more than a mathematical fellow traveler (though he might deny this, being famous for his modesty). To wit, with obvious warm friendship for the community he sprang from, he as recently as January 2009 contributed a thought provoking article to Notices of the AMS on “frogs” and “birds.” This was not a foray into biology: roughly, birds are mathematical architects looking for far flung structural connections (or maybe not so far flung, I guess), and frogs are diggers and problem solvers who prefer to cultivate their own part of the mathematical garden. Taking the kind of physics Dyson does as ultimately part of mathematics, Freeman Dyson, qua mathematician, exemplifies that rara avis that flies across widely spread landscapes, taking in number theory, quantum electrodynamics, and even the psychology or sociology of mathematicians. It is this polymathy on Dyson’s part that I want to propose as a prelude to what Discrete Groups, Expanding Graphs and Invariant Measures presents to us. It is truly mathematics for the birds (if I may put it so egregiously) and requires a Dysonian mindset for its full appreciation.

The Ramanujan conjecture figures hugely in this book, and Dyson, who as a Cambridge undergraduate proposed the idea of a “crank” in connection with the Ramanujan congruences for the partition function, would be drawn like a bee to honey to this kind of connection betwixt wildly disparate areas. In fact, this bit of biography amplifies Dyson’s fittingness as the exemplar of the type of mathematical scholar the book under review would most appeal to: after all his number theoretic work is characterized by a marvelous inventiveness trained at the solving of problems, and the subject matter of Discrete Groups, Expanding Graphs and Invariant Measures is undeniably schizoid in this sense.

It is the main thrust of Discrete Groups, Expanding Graphs and Invariant Measures that the Ramanujan conjecture, more precisely Ramanujan-Petersson as established by Deligne, and Kazhdan’s property (T) from the representation theory of semi-simple Lie groups play a huge role in the solution of a pair of problems, one in the theory of expanding graphs, which really belongs to computer science, the other in the theory of invariant measures, specifically an old question about Lebesgue measure on subsets of the n-sphere. Here is what the author of the book under review says:

In the last fifteen years two seemingly unrelated problems … were solved by amazingly similar techniques [!] from representation theory and from analytic number theory. One problem is the explicit construction of expanding graphs … [being] basic building blocks for various distributed networks, [while t]he other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [asking] whether the Lebesgue measure is the only finitely additive measure of total measure one, defined on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations.

Evidently there is a lot of mysterious and beautiful mathematics hiding in the shadows: this calls for a very high-flying bird, indeed.

Taking first the (historically prior) representation theoretic connection, the major token in the game is Kazhdan’s “property (T),” in the context of semi-simple Lie groups, and the major players in the game are Grigory Margulis vis à vis the expanding graphs problem, whereas the solution of the Ruziewicz problem using property (T) occurred at the hands of Rosenblatt, Margulis, Vladimir Drin’feld, and Dennis Sullivan. On p. 20 of Discrete Groups, Expanding Graphs and Invariant Measures we learn that Kazhdan’s property (T) for a given group G entails that “every unitary representation which has almost invariant vectors contains also an invariant vector,” where the notion of an almost invariant (unimodular) vector x for a unitary representation F in some Hilbert space has to do with F(x) – x being is some sense small on compacta.

And then the (historically second) application of the Ramanujan conjecture to the matter under consideration involved Margulis again, as well as the author, Lubotzky, himself, together with R. Phillips and Peter Sarnak (making for another connection with Freeman Dyson, as hinted above). In this connection, recall that the Ramanujan conjecture states that the Fourier coefficients of the weight 12 Weierstrass cusp form ∆(z), evaluated at the rational primes, p, are bounded by 2p11/2, a fact that follows from Deligne’s proof of the Weil conjectures (so Ramanujan’s conjecture is a theorem). This jewel of number theory was obtained as a consequence of Deligne’s magnificent manoeuvres with Grothendieck’s machinery which was originally designed for algebraic geometry.

On p. 77 of Discrete Groups, Expanding Graphs and Invariant Measures Lubotzky notes that the result needed “for both the final solution of the Banach-Ruziewicz problem and the construction of Ramanujan graphs … [is] Deligne’s Theorem … which is, in fact, a representation theoretic reformulation of [the] Ramanujan conjecture (known also as Petersson’s conjecture) …” He goes on to say that “both problems can be solved using some weaker results proved earlier by Rankin and Eichler, respectively, but Deligne’s Theorem gives a unified approach as well as better results in the Ruziewicz problem…”

The book under review won the Ferran Sunyer i Balaguer Award and this makes for high expectations. We are not disappointed: ten chapters beautifully arranged and covering introductory material, methodology, explication of sophisticated techniques, and then resolutions of the problems in question by the two methods given. Beyond this there is a section on open problems (perhaps dated by now) and an appendix by Jonathan Rogawski which is itself worth the price of admission (says Math Revs, cited on the back cover: “The Appendix, written by J. Rogawski, explains the Jacquet-Langlands theory and indicates Deligne’s proof of the Petersson-Ramanujan conjecture. It would merit its own review …”)

And the mathematics in Discrete Groups, Expanding Graphs and Invariant Measures is nothing short of delectable, including an explication of invariant measures starting with the Banach-Tarski paradox, a crisp account of the representation theory surrounding Kazhdan’s property (T) leading already on p. 30 ff. to a solution à la Margulis of the expander construction problem — existence is “easy” (see p. 5), and leading to a solution of the (Banach-) Ruziewicz problem in dimensions ≥ 4 already on p. 34 ff. This wonderful first course, so to speak, is then followed by a discussion of three of the most important themes in modern number theory (read: automorphic forms and representations, etc.), namely, the eigenvalues of the Laplacian for a suitable Riemannian manifold (and Lubotzky takes us from the geometric Laplacian to the combinatorial Laplacian), the representation theory of the projective general linear group, and then the spectral decomposition of attendant Hilbert spaces, typically on adèlic symmetric spaces. Here, not surprisingly, Atle Selberg’s work figures very prominently in Lubotzky’s account, as does Langlands’. And at this point in the proceedings we reach the themes of improving the solution to the Banach-Ruziewicz problem: dimensions 2 and 3 are covered (the case originally dealt with by Drin’feld), and then Ramanujan graphs take the stage in a very big way — and pretty much stay there.

The pedagogy of Discrete Groups, Expanding Graphs and Invariant Measures is very good, too, in that Lubotzky uses the time-honored speech-maker’s approach, “tell ‘em what you’re going to tell ‘em; tell ‘em; then tell ‘em what you told ‘em,” very effectively. He introduces the reader to a lot of very deep mathematics in a smooth and accessible manner, next he does the mathematics clearly and concisely (this book is not for the timid!), and then he returns to the material with further explanation, appraisal, and prophecy.

It is no wonder that Discrete Groups, Expanding Graphs and Invariant Measures won the Sunyer i Belaguer prize: it is a gem.


Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Wednesday, December 9, 2009
Reviewable: 
Yes
Include In BLL Rating: 
No
A. Lubotzky
Series: 
Modern Birkhäuser Classics
Publication Date: 
2010
Format: 
Paperback
Category: 
Monograph
Michael Berg
06/21/2010
Publish Book: 
Modify Date: 
Monday, June 21, 2010

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