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).



What are the best settings for the large scale geometry of locally compact groups?
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.


