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

[摘要] For a topological group G, the classifying space BG is an aspherical CW-complexsuch that its fundamental group is G and the universal cover is contractible. In thealgebraic K- and L-theories, many long-standing conjectures use notions of the classifyingspace 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, manyof 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 compactificationprovides finite models of the proper classifying space for arithmetic groups insemisimple Lie groups. Margulis’s arithmeticity theorem implies that the Borel-Serreparital compactification also provides finite models for irreducible lattices in higherrank semisimple Lie groups. The existence of finite models of the proper classifyingspace 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 classifyingspace for general lattices in a semisimple Lie group of R-rank one. This isa generalization of the Borel-Serre partial compatification of rank one symmetricspaces. The resulting finite model is a manifold with boundary and it is obtainedfrom the rank one symmetric space by attaching geometrically rational boundarycomponents with respect to the lattice in concern. The main tools used in the proofare the topology defined by convergence class of sequences, the continuous and properaction of the lattice extended from the canonical action on the interior, Garland andRaghunathan’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 subanalyticG-manifolds.

[发布机构] University of Michigan

[效力级别] Lattices in Semisimple Lie Groups

[关键词] Proper Classifying Space;Lattices in Semisimple Lie Groups;Borel-Serre Compactification;Symmetric Spaces;Mathematics;Science;Mathematics