# Geometric group theory and analysis

Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. Is there any subfield of GGT where methods from analysis find applications? I once heard that analytical tools, e. g. geodesic flows, are used in studying ends of groups, but that's all I know.

Analytic ideas enter into several parts of geometric group theory. Tom already mentioned amenability, so I'll skip that.

1) Complex analysis and the theory of quasiconformal mappings plays an important role in understanding the mapping class group, which is one of the most important groups studied by geometric group theorists. For information on this, see Farb and Margalit's forthcoming book "A primer on mapping class groups", available on either of their webpages.

2) The study of groups with property (T) ends up using quite a bit of analysis. See the book "Kazhdan's Property (T)" by de la Harpe, Bekka, and Valette for information on this.

3) Analysis (together with ideas from ergodic theory, which is of course quite analytic) plays an important role in proving various rigidity theorems. The most famous is the Mostow Rigidity Theorem, whose original proof uses lots of analysis : quasiconformal mapping in high dimensions, the fact that Lipschitz functions are differentiable almost everywhere, ergodic theory, etc.

I'll expand a little on Andy and Tom's very nice answers.

Property (T), amenability and rigidity theorems are not really separate topics. For a beautiful introduction, including some of the connections between these and other topics, see Lubotzky's book "Discrete Groups, Expanding Graphs, and Invariant Measures". Another great reference is Zimmer's book "Ergodic Theory and Semisimple Groups".

As for whether this sort of stuff counts as a "subfield" of geometric group theory, that probably depends on your definition of geometric group theory.

• Hi Anne! I've always thought that geometric group theory was less a subject than an ideology (like communism). Almost any area of mathematics that involves algebra or geometry/topology can be viewed through a geometric group theory lens. The best definition of geometric group theory is thus "the kinds of things that people who call themselves geometric group theorists study". But perhaps I only claim that so I can call what I do geometric group theory... Mar 14, 2010 at 15:46
• Andy - what does the geometric-group-theory lens tell us about algebraic geometry, then? Really, I'd love to know!
– HJRW
Mar 14, 2010 at 18:35
• Well, do you count Margulis's arithmeticity theorem as geometric group theory? Really, the whole theory of lattices in Lie groups is a meeting group for ideas from algebraic geometry and geometric group theory. In a different direction, there is Lars Louder's work on the "algebraic geometry" of limit groups. Finally, ideas from geometric group theory play an important role in understanding the geometry/topology of the moduli space of curves. Mar 14, 2010 at 18:49
• Andy: the 'algebraic geometry' of limit groups really is due originally to people like Sela and Kharlampovich--Miasnikov, (Lars and the rest of us came along later) and has very little to do with actual algebraic geometry. The other examples are OKish, but really only work because there's a (discrete) group floating around. At the risk of putting words into Anne's mouth: I think her point was that there are lots of people who study rigidity etc who might not call themselves geometric group theorists. Some might call themselves asymptotic group theorists, for instance.
– HJRW
Mar 18, 2010 at 17:26

One point of contact between GGT and analysis is the study of amenability. See for instance the wikipedia article on amenable groups. As you can read there, there are many equivalent conditions for a (discrete) group to be amenable, some couched in terms of analytic concepts. On the other hand, if you have a look at the usual proof that the free group on two generators is not amenable, that has a definite flavour of geometric group theory about it.

(There is also the saga of whether Thompson's group $F$ is amenable. Danny Calegari has an excellent blog post on this. In short, the situation is this. For many years, the question of whether $F$ is amenable has been an open question. But in 2009, one paper appeared purporting to prove that $F$ is amenable, and another paper appeared purporting to prove that $F$ is not amenable. The one claiming the positive result uses a kind of analysis rarely seen in GGT, and so received a lot of attention. As far as I can tell, both authors have now admitted serious flaws in their papers, but neither has withdrawn it from the arXiv.)

• As far as I understand arXiv policy, you can't withdraw a paper from arXiv – though I suppose one could post an updated version explaining that there is a problem and what it is. Mar 14, 2010 at 14:36
• There is an action you can take on the arxiv called withdrawing your paper. Functionally, it is equivalent to uploading a new, empty, version of your paper, along with the comment "This paper has been withdrawn." But what you say is in some sense true in that you cannot remove old versions of submissions on the arxiv. Mar 14, 2010 at 14:49
• Right, Harald, I was being lazy there. I meant what Pete said. If you google "withdrawn site:arxiv.org" you'll see that this procedure has been used plenty of times. Mar 14, 2010 at 15:12

The work of Yu, partly joint with Kasparov, connects geometric group theory to the geometry of Banach spaces: If a finitely generated group embeds coarsely (in the sense of Gromov) into a uniformly convex Banach space, then it satisfies the coarse Baum-Connes conjecture. In effect, this means you can work on certain problems in geometric group theory without knowing the definition of a group (e.g. by determining when a metric space with bounded geometry coarsely embeds into a uniformly convex Banach space).

Not sure why this is not being mentioned but one prominent use of analysis in geometric group theory is in studying boundaries of groups (towards Cannon's conjecture and beyond). See for example discussion here and here.