lattices in semisimple Lie groups I would like to learn more on lattices in semisimple Lie groups, especially their relations with Coxeter groups. Does anyone have suggestions of books that could be useful?
Thanks!
 A: I realize that this is a very old post, still here is an answer:


*

*If you want to understand (connected) semisimple Lie groups, then indeed you have to learn a bit about finite Coxeter groups first, since they are closely tied to the root systems. For that, say, Bourbaki's "Lie Groups and Lie Algebras" or Humphreys "Introduction to Lie algebras and representation theory" are good references (there are many other sources).

*If you want to understand lattices in semisimple Lie groups, then, indirectly, you see their relation to finite Coxeter groups. There are several books dealing with lattices in semisimple Lie groups, the gentlest one, I think, is Introduction to Arithmetic Groups by Dave Witte Morris. You can also read Zimmer's book Ergodic theory and semisimple groups or Raghunathan's book 
"Discrete subgroups of Lie groups" (both are still reasonably gentle); the hardest one to read will be the most important one, by Margulis
"Discrete Subgroups of Semisimple Lie Groups" mentioned by Bill Cook in his comment. 

*After spending few years going through these books you may suddenly remember that you were interested in lattices in connection to Coxeter groups and you find almost nothing about Coxeter groups in these sources. This is due to the following:
Theorem. If $\Gamma$ is a Coxeter group isomorphic to a lattice in a simple noncompact Lie group $G$, then $G$ is (locally) isomorphic to $O(n,1)$ for some $n$. (In fact, you can replace simple with semisimple, then $G$ will be locally isomorphic to a product of $O(n_i,1)$'s.) 
Proof. Unless $G$ is locally isomorphic to $SO(n,1)$ or $SU(n,1)$, it has Kazhdan's Property T (whatever it is), hence, every lattice in $G$ has property T, but infinite finitely generated Coxeter  groups never have Property T. Lastly, lattices in $SU(n,1)$ were shown not to be isomorphic to Coxeter groups in this paper by Pierre Py (actually, he proves more). qed 
Thus, there is almost no connection between Coxeter groups and lattices except for a negative one. That is, except for the fact that many nice examples of lattices in $O(n,1)$ (for small $n$) are indeed  given by Coxeter groups. See for instance my answer to this MSE question. Well, actually, there is more to this story, since, according to theorems by Vinberg and Prokhorov, there are no reflection subgroups in $O(n,1)$ which are lattices, provided that $n\ge 997$. Thus, most likely, groups $O(n,1)$, $n\ge 1000$, contain no lattices which are isomorphic to Coxeter groups.  
