What are the best settings for the large scale geometry of locally compact groups? 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).
 A: 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.
A: 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).
A: 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.
A: Maybe the following reference is of value:


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*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)

A: 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:


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*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.

*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.
