# 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!

• I suggest Siegel's "Lectures on the geometry of numbers", lectures XIV and XV Dec 5, 2011 at 20:16
• Borel's softcover book Introduction aux groupes arithmetiques (Hermann, 1969) is helpful for fundamental domains (though it still has notational inconsistencies after he spliced two older sets of lecture notes). For some special cases I tried to write this down in Borel's style but in English in my Springer Lecture Notes 789 Arithmetic Groups. In general, the volume of a fundamental domain will be infinite. Dec 5, 2011 at 23:34
• About infinite volume (as raised by @Jim H.): this is a genuine issue for "Fuchsian" subgroups of SL(2,R), and more generally for real-rank-one groups to which (Mostow-Margulis-Zimmer) rigidity does not apply. One could already declare that one's only interest was in arithmetic subgroups (congruence-or-not), in which case volumes are always finite. Nevertheless, indeed, there is a substantial school of investigation concerning non-finite-co-volume discrete subgroups of SL(2,R), SL(2,C), and SO(n,1)'s more generally. (I don't know/understand the sensibilities/goals of that.) Dec 5, 2011 at 23:50