Geodesic
Volumes of symmetric spaces via lattice points
Documenta mathematica, Tome 11 (2006), pp. 425-447
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We show how to use elementary methods to compute the volume of Slk​R/Slk​Z. We compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. We also present a proof of the closely related result that the Tamagawa number is 1.
DOI : 10.4171/dm/217
Classification : 11F06, 11H06, 11M45
Mots-clés : volume, arithmetic group, lattice, special linear group, Tauberian theorem, Tamagawa number 
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     author = {Henri Gillet and Daniel R. Grayson},
     title = {Volumes of symmetric spaces via lattice points},
     journal = {Documenta mathematica},
     pages = {425--447},
     year = {2006},
     volume = {11},
     doi = {10.4171/dm/217},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/217/}
}
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Henri Gillet; Daniel R. Grayson. Volumes of symmetric spaces via lattice points. Documenta mathematica, Tome 11 (2006), pp. 425-447. doi: 10.4171/dm/217

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