The congruence subgroup kernel and the reductive Borel-Serre compactification. Algebraic Geometry and Number Theory seminar at John Hopkins University, Baltimore. March 1, 2011
Invited Lectures ; Let G be a reductive algebraic group defined over a number field. The congruence subgroup kernel quantifies to what extent is every arithmetic subgroup of G a congruence subgroup. It has been studied extensively. On the other hand, the reductive Borel-Serre compactification of an arithmetic quotient of the symmetric space associated to G reflects the geometry of this quotient at infinity. Its cohomology has been extensively studied in view of applications to automorphic forms. After describing these two seemingly disparate objects I will show how the congruence subgroup kernel can be related to the fundamental group of the reductive Borel-Serre compactification. There is a generalization to S-arithmetic subgroups. This is joint work with Lizhen Ji, V. Kumar Murty, and John Scherk.