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My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still meaningful and valid in the large scale geometry (or coarse geometry) of locally compact groups. But I don't even know if there is a comprehensive theory for large scale geometry of locally compact groups. I have not been able to find any substantial resource in the literature, except a paper by Udo Baumgartner titled "Totally disconnected locally compact groups as geometric objects". Unfortunately, this paper does not try to create a theory parallel to geometric group theory of finitely generated groups. On the other hand, although as soon as we have a metric on a locally compact group we can define and study its coarse geometry, but I am not aware of any general method for defining metrics on locally compact groups. So it seems studying the actions of locally compact groups on "nice metric spaces" would be the best place to start discovering the geometric features of locally compact groups. But, again, I need a good place to start!

So, I hope someone can help me to find some resources to gather information about the large scale geometry of locally compact groups (if there exists any).

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You might be interested in a Yves Cornulier's survey on compactly presented groups. arxiv.org/abs/1003.3959 –  David Cohen Jun 13 '13 at 17:19
    
If you need a metric: On a locally compact group you can choose a left-invariant, complete metric such that every bounded set is relatively compact if and only if the group is second-countable. Especially, a locally compact group is a Polish group if and only if it is second-countable. –  The User Jun 13 '13 at 17:22
    
@TheUser: I appreciate if you can introduce a reference for this statement. Because I am also interested to continuous length functions on locally compact groups and every left invariant metric gives me one. –  Vahid Shirbisheh Jun 13 '13 at 17:29
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Thanks for pointing to these notes. Here are a considerably extended version of these notes (book in preparation with Pierre de la Harpe) normalesup.org/~cornulier/MetricLC.pdf –  YCor Jun 13 '13 at 18:40
    
@YvesCornulier: Thank you very much for sharing your unfinished work. –  Vahid Shirbisheh Jun 13 '13 at 18:47

5 Answers 5

up vote 7 down vote accepted

I can give an example from my own papers which uses these ideas, a result of myself, Sageev, and Whyte in our paper "Quasi-actions on trees I: Bounded valence", MR1998479. This is a rigidity theorem for locally compact groups $G$ that contain a discrete cocompact finite rank free subgroup: $G$ must act continuously, properly, and cocompactly with compact kernel on some tree $T$ of uniformly bounded valence.

One phrase that might help you find further information in the literature is ``compactly generated''. For instance, it is kind of a folk theorem that the Milnor-Svarc theorem works for a compactly generated group: the word metrics associated to compact generating sets are all quasi-isometrically related to each other via the identity map, and more generally any topologically reasonable action of the group on a proper geodesic metric space induces an equivariant quasi-isometry between the group and the metric space. Examples of this include, of course, actions of semisimple Lie groups on their associated symmetric spaces.

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And I see that my answer has crossed with David's pointer to Yves Cornulier's paper. –  Lee Mosher Jun 13 '13 at 17:39

Let me extend my comment above to give some more detailed information, especially regarding metrisability (I am not aware of any “theory: First of all it is well-known that a topological group is metrisable if and only if it is first-countable. However, such a metric might not have nice properties. However, a locally compact group is metrisable by a left-invariant, proper (every bounded closed set is compact) metric if and only if the group is second-countable, see the papers Metrics in locally compact groups and Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces. Properness of the metric implies completeness, thus every second-countable locally compact group is actually a Polish group.

Actions of Polish groups have been studied extensively in descriptive set theory, especially continuous actions on Polish (completely metrisable) spaces, but as usual in descriptive set theory also cases of measurable actions with respect to some $\sigma$-algebras. Gregory Hjorth has written a book Classification and Orbit Equivalence Relations and a shorter chapter A survey of current and recent work on the theory of Borel equivalence relations for the Handbook on Set Theory, which appeared recently (this topic has been applied to C*-algebras and dynamical system theory, too, I am no expert, but I am mentioning it since you mention these topics in your profile). Locally compact Polish groups are kinda the simplest case in this theory, local compactness guarantees that the group actions are not “turbulent”, their orbits are $\Sigma^0_2$-sets (countable unions of closed sets). I do not know what you do expect exactly, maybe this theory is useless for you, but it might be reasonable to notice its existence. Invariant Descriptive Set Theory by Su Gao is another recent monograph covering Polish group actions extensively. Let me quote a result from that theory that might give you some impresion of what is known about the general structure of such group actions:

Given an action of a locally compact group acting continuously on a Polish space $X$. Then there exists a Polish space $Y$, a Borel measurable equivalence relation $E\subset Y\times Y$ such that every equivalence class is countable and a Borel measurable function $\theta\colon X\to Y$ such that for all $a,b\in X$ we have $Ga=Gb\Leftrightarrow \theta(a)E \theta(b)$.

Using a web search I have also found this document considering left-invariant, proper metrics on locally compact groups which do not generate the topology. Under that condition you can also consider some non-second-countable groups (sorry, I did not notice the comment which already referenced this).

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Thank you very much for the references. –  Vahid Shirbisheh Jun 13 '13 at 21:00
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Indeed as far as large-scale geometry is concerned, the good object to consider on a locally compact group is a pseudo-metric for which the bounded subsets are exactly the subsets with a compact closure (for the original topology, insofar as we don't consider the topology defined by the pseudo-metric). Example: the word length w.r.t. a compact generating set. –  YCor Jun 13 '13 at 21:22
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If you want to prove that locally compact second countable groups are Polish, you don't need anything nearly as fancy: A locally compact second countable space is metrizable and open in any of its completions, so it is Polish. –  Martin Jul 2 '13 at 12:17

I just want to introduce a paper which is closely related to my question and might be useful to people interested in the subject:

Krön, Bernhard, and Rögnvaldur G. Möller. "Analogues of Cayley graphs for topological groups." Mathematische Zeitschrift 258.3 (2008): 637-675.

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Yes, this paper is very well-written. It associates to every compactly generated locally compact group a connected finite valency graph with a proper vertex-transitive action (now known as "Cayley-Abels graph"). This was originally introduced by Herbert Abels in his 1974 paper (in German) on ends of locally compact groups. –  YCor Jul 2 '13 at 11:56

This question has already been answered, but I thought it would be a good idea to draw attention to some things that have appeared since the question was asked:

  1. There is an upcoming book by Yves de Cornulier and Pierre de la Harpe called 'Metric geometry of locally compact groups'. I don't think it has been published yet, but there is a preliminary version on the arXiv that is well worth browsing.

  2. Christian Rosendal has put some papers on the arXiv concerning the large-scale geometry of metrisable groups. I think these take more of a model-theoretic perspective.

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The first item has already been discussed in the comments. –  Vahid Shirbisheh Jul 7 at 10:54

Maybe the following reference is of value:

  • MR2337107 Reviewed Hofmann, Karl H.; Morris, Sidney A. The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. EMS Tracts in Mathematics, 2. European Mathematical Society (EMS), Zürich, 2007. xvi+678 pp. ISBN: 978-3-03719-032-6 (Reviewer: Markus Stroppel)
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This paper concerns the structural theory of connected locally compact groups, but I don't think it deals with the large scale geometry. –  YCor Jul 2 '13 at 14:29

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