Cut and Paste Invariants and Duality: A Motivating Example Via zeta(-1), zeta(2) and SL_2Z

Here’s something enticing and strange: there are two “cut and paste” invariants of the same group which are equal to dual zeta values! the euler characteristic \( \chi(SL_2(\mathbb{Z})) = \zeta(-1), \) and the tamagawa measure of \( \mu(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}) = \zeta(2)\).