Metric Geometry of Locally Compact Groups

It can probably be argued pretty easily that the best of all possible groups are Lie groups. After all, what more can one wish for than to be able to do full-fledged (local) analysis on a group, and then to get a natural association to an algebra, the corresponding Lie algebra, which is a marvel in its own right.

Along the lines of “What came first, the chicken or the egg?,” we might ask which is more fundamental, the Lie group or the Lie algebra? It really does depend on your point of view: in the study of differentiable manifolds (very natural objects, no?) the Lie algebra of derivations comes first, and that’s where my sympathies lie, I guess (and that’s no lie, and not a Lie either: let’s get that pun out of the way immediately). On the other hand, Sophus Lie himself looked at what are now called analytic groups first, I think, so history comes down on that side. In any case, Lie group structure is certainly a consummation devoutly to be wished, to quote Shakespeare (Hamlet, Act III, Scene I). But it ain’t necessarily so, to rebut with the Gershwin Brothers (Porgy and Bess, Act 2, Scene 2): many times throughout mathematics, we miss out on getting Lie structure, and have to settle for less. Often it’s not all that bad a deal, however, since we get other flavors of analytic structure on certain well represented groups (and yes, I know, it’s another pretty heinous pun).

So let’s cut to the chase as far as the book under review goes: locally compact groups are wonderful things in themselves, and in and of themselves provide a very fecund interplay between their algebraic and topological qualities. In my own experience I have dealt with LCA (locally compact abelian) groups explicity — for example, I sport the book, The Structure of Locally Compact Abelian Groups, by Armacost, on my shelves — and I have dealt with Fourier analysis on locally compact groups a bit (cf. the wonderful book, Introduction to the Representation Theory of Compact and Locally Compact Groups, by Alain Robert). In other words, in my experience, it’s been about structure theory and (generalized?) harmonic analysis; in fact, there’s the prominent example, Fourier Analysis on Groups, by, yes, Walter Rudin. How does all this stack up against what we have on the table now?

Well, it’s a different dish, really. Cornulier and de la Harpe focus on discrete groups as metric spaces (the first topic in their Introduction, in fact), and state the following: “Whenever a group \(\Gamma\) appears in geometry, which typically means that \(\Gamma\) acts on a metric space of some sort…, the geometry of the space reflects some geometry of the group.” They mention a pretty large number of examples of what they mean here, among which we just mention two: “’Dehn Gruppenbild’ (also known as Cayley graphs), used to picture finitely generated groups and their word metrics, in particular knot groups…” and “…the tightly interwoven developments of combinatorial group theory and low dimensional topology, from Dehn to Thurston, and so many others.” And then we read: “From 1980 onwards, … under the guidance of [Mikhail] Gromov, … the group community has been used to consider a group (with appropriate conditions) as a metric space, and to concentrate on large-scale properties of such metric spaces.”

In this context the authors mention discrete groups, countable groups, finitely generated groups, and finitely presented groups: “each class properly contain[s] the next one” so we’re always dealing with discrete groups here. And note that we have some famous examples already, e.g. every modular former’s favorite, \(\mathrm{PSL}(2,\mathbb{Z})\).

But that’s just the start of the story. Next, Cornulier and de Harpe state that “[i]t has long been known that the study of a [discrete] group \(\Gamma\) can be eased when it sits as a discrete subgroup of some kind of locally compact group \(G\): again something we number theorists (full disclosure, I guess …) are very familiar with. But the authors get more specific, if austere (to the point of exceeding their book’s charter): “The following two standard examples, beyond the scope of the present book, involve a lattice \(\Gamma\) in a locally compact group \(G\): first, [the] Kazhdan property is inherited from \(G\) to \(\Gamma\); second, if \(\Gamma\) is moreover cocompact in \(G\), cohomological properties of \(\Gamma\) can be deduced from information on \(G\) or on its homogeneous spaces.” Zowie, or, as my teenage sons are apt to say, “pretty sick!” In any case, you begin to get a feel, I hope, for what we’re dealing with here: this is some very serious stuff.

So here’s what the authors are up to: “Chapter 2 contains fundamental facts on LC-spaces and groups … Chapter 3 deals with two categories of pseudo-metric spaces that play a major role in our setting … Chapter 4 shows how the metric notions of Chapter 3 apply to LC-groups … Chapter 5 contains essentially examples of compactly generated LC-groups … Chapter 6 deals with the appropriate notion of simple connectedness for pseudo-metric spaces … Chapter 8 [addresses the fact that] an LC group is compactly presented if and only if the [associated] pseudo-metric space … is coarsely simply connected …” Happily there are examples of the latter otherwise dauntingly exotic beasts; again I’ll mention just two: “abelian and nilpotent compactly generated LC-groups … [and] … \(\mathrm{SL}_n(\mathbb{K})\), for every \(n\geq 2\) and every local field \(\mathbb{K}\),” and with this last bit we get our feet on the ground again, if only for a moment: again, this is bread and butter for (us) number theorists and, indeed, it’s part of the menu for just about everybody else.

So, again, this is very serious stuff. The back-cover says that “[t]he book is aimed at graduate students and advanced undergraduate students, as well as mathematicians who wish some introduction to coarse geometry and locally compact groups.” Allow me to reply with, respectively, “fair enough,” “not so much,” and “yes, all right.” The book is certainly pedagogically on target: lots of examples, careful proofs, a very thorough treatment — the whole nine yards. But it’s only fair to note that this is specialized and austere material, and the reader should not only be well-prepared (mathematically, and also psychologically) but be pretty committed to learning this material. I’ve said it before in other contexts, but it certainly applies now: not for the timid.

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