Reference request: The geometry of $GL_2(\mathbb{R})$ and related questions Can anyone please recommend some good reading on the geometry of linear groups and their actions?
An example of the kind of question I am interested in: Explicitly describe a fundamental domain for the action of $GL_2(\mathbb{Z})$ on $GL_2(\mathbb{R})$, and compute the volume of the quotient. 
I'm familiar with this particular question and its answer, but it is evidently a special case of a more general theory and I would love to see it treated in context. I looked briefly at Borel's Linear Algebraic Groups, Lang's $SL_2(\mathbb{R})$, a couple intro books on Lie groups -- and at a brief glance, none of them seemed to squarely address this kind of question.
Thank you!
 A: In addition to Siegel's "Geometry of Numbers", Godement's Seminaire Bourbaki from 1967-8 does "reduction theory" including a very nice adelic version of Minkowski-Siegel-Borel.
If the goal is obtaining a Siegel-set approximation to a fundamental domain, rather than a precise fundamental domain, the GL(n,Q) case (and the anisotropic orthogonal group case...) is recapped in http://www.math.umn.edu/~garrett/m/v/reduction.pdf
A: You can try to take a look in Dave Witte's book about arithmetic groups here - http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html
He also presents in his site a dynamical approach to this subject (due to Margulis) which proves that sets like G(Z) are lattices in fairly general settings, without the full reduction theory - http://people.uleth.ca/~dave.morris/talks/arith-grps-are-latts-chgo-6-10.pdf.
The volume computations is easy when you have explicit fundemental domain, but as Paul mentioned, you can usually get only a Siegel domain.
The general theory here is due to Langlands via the theory of Eisenstein series. You can look here - http://publications.ias.edu/sites/default/files/chev-ps.pdf.
If you want to see just more explicit formula (say for PSL2), it appears as a guided exercise in Bump's book.
Another approach is to use the Siegel-Weil theorem, as Alex Eskin indicates here - Volume of fundamental domain and Haar measure
