Volumes of symmetric spaces via lattice points

Abstract

In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can 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 of Sl_k Q is 1 that is somewhat simpler and more arithmetic than Weil's. His proof proceeds by induction on k and appeals to the Poisson summation formula, whereas the proof here brings to the forefront local versions of the formula, one for each prime p, which help to illuminate the appearance of values of zeta functions in formulas for volumes.