Cofinite Classifying Spaces for Lattices in R-Rank One Semisimple Lie Groups.

Abstract

For a topological group G, the classifying space BG is an aspherical CW-complex such that its fundamental group is G and the universal cover is contractible. In the algebraic K- and L-theories, many long-standing conjectures use notions of the classifying space and the classifying spaces for various families of subgroups. Therefore, finding concrete and simple models of the classifying spaces is an important subject.

The proper classifying space is the classifying space for the family of finite subgroups. There are many finite dimensional models for proper classifying spaces, many of which are finite, meaning that they consist of only finitely many equivariant cells. As observed by Adem and Ruan and proved by Ji, the Borel-Serre partial compactification provides finite models of the proper classifying space for arithmetic groups in semisimple Lie groups. Margulis’s arithmeticity theorem implies that the Borel-Serre parital compactification also provides finite models for irreducible lattices in higher rank semisimple Lie groups. The existence of finite models of the proper classifying space for general lattices in semisimple Lie groups is not yet fully known until now.

The thesis presents the method of constructing finite models of the proper classifying space for general lattices in a semisimple Lie group of R-rank one. This is a generalization of the Borel-Serre partial compatification of rank one symmetric spaces. The resulting finite model is a manifold with boundary and it is obtained from the rank one symmetric space by attaching geometrically rational boundary components with respect to the lattice in concern. The main tools used in the proof are the topology defined by convergence class of sequences, the continuous and proper action of the lattice extended from the canonical action on the interior, Garland and Raghunathan’s reduction theory on lattices in semisimple Lie groups of R-rank one, and Illman’s theorem on the existence of G-CW-complex structures on subanalytic G-manifolds.