Tuesday February 1, 2005
2:30 pm, Wheatley 2-0042

Eliot Brenner

Yale University

Fundamental Domains for the Action of Arithmetic Groups on Symmetric Spaces

Abstract: We study a type of exact fundamental domain $F$ for the action of an arithmetic group $\Gamma$ on a symmetric space of non-compact type that is more symmetric and better suited for the study of Eisenstein series and spectral analysis than the standard "nearest neighbors" or "Dirichlet" domain. The construction is based on a modification of the Dirichlet construction that takes advantage of the properties of the action $\Gamma$ with respect to the standard Iwasawa coordinates. The demand for a more symmetric domain comes from the ongoing project of Jorgenson and Lang on heat kernels and explicit spectral analysis on arithmetic quotients of symmetric spaces. In order to keep the exposition as down-to-earth as possible, we will focus mainly on the case of $\Gamma = SL(2,Z[i])$, in which case $F$ reduces to the standard "Picard domain." In this case, we will give a complete proof of the first main result concerning the symmetry of $F$, namely that $\Gamma$ exactly tiles the fundamental domain of $\Gamma_U$ with translates of $F$. If time permits, we will indicate how the construction can be generalized to the case of $SL(n)$ for larger $n$, in which case the domains $F$ are similar to those previously considered by Douglas Grenier, and to other reductive groups, in which case they represent a new construction.