# 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!

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I suggest Siegel's "Lectures on the geometry of numbers", lectures XIV and XV –  Alain Valette Dec 5 '11 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. –  Jim Humphreys Dec 5 '11 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.) –  paul garrett Dec 5 '11 at 23:50
thanks to all ! –  Frank Thorne Dec 15 '11 at 0:46

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

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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

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