Volumes of Symmetric Spaces via Lattice Points

We show how to use elementary methods to compute the volume of $Sl_k R/Sl_k Z$. 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 $1$.

2000 Mathematics Subject Classification: 11F06, 11H06, 11M45.

Keywords and Phrases: special linear group, volume, lattice, Tauberian theorem, arithmetic group, Tamagawa number.

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