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)

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/0605013

30 pages. Article gives summary of main results of the e-print math.NT/0605012, with addition of background information and motivation. Submitted to the Israel Journal of Mathematics

arxiv created 2006/11/22 · arxiv updated 2009/12/01

Abstract

Let G=SO(3,C), Γ=SO(3,Z[i]), K=SO(3), and let X be the locally symmetric space Γ\backslash G/K. In this paper, 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 Γ_Z=SO(2,1)_Z of Γ acting on the symmetric space G_R/K_R, where G_R is the split real form SO(2,1) of G and K_R is its maximal compact subgroup SO(2). We formulate a simple geometric relation between the fundamental domain of Γ and Γ_Z so described. These fundamental domains are geared towards the detailed study of the spectral theory of X and the embedded subspace X_R=Γ_Z\backslash G_R/K_R.

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