2006/04/29 by Eliot Brenner, Brenner, Eliot
Mathematics · #11F55 (Primary) #11F72 #11H55 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F55 #msc:11F72 #msc:11H55
paper · pdf · doi:10.48550/arxiv.math/0605012
119+ii Pages, 1 Figure, contains proofs of main results in "A fundamental domain of Ford type for $SO_3(Z[i])\backslash SO_3(C)/SO(3)$ and for $SO(2,1)_Z\backslash SO(2,1)/SO(2)$"
arxiv created 2006/04/29 · arxiv updated 2009/12/01
We initiate a study of the spectral theory of the locally symmetric space X=Γ\backslash G/K, where G=SO(3,Complex), Γ=SO(3,Z[i]), K=SO3. We write down explicit equations defining a fundamental domain for the action of Γ on G/K. The fundamental domain is well-adapted for studying the theory of Γ-invariant functions on G/K. We write down equations defining a fundamental domain for the subgroup ΓZ=\SO(2,1)Z of Γ acting on the symmetric space GR/KR, where GR is the split real form \SO(2,1) of G and KR is its maximal compact subgroup \SO(2). We formulate a simple geometric relation between the fundamental domains of Γ and ΓZ so described. We then use the previous results compute the covolumes of of the lattices Γ and ΓZ in G and GR.