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THE LARGE SCALE TOPOLOGY AND ALGEBRAIC K-THEORY OF ARITHMETIC GROUPS

By Boris Goldfarb

Abstract

Abstract. New compactifications of symmetric spaces of noncompact type X are constructed using the asymptotic geometry of the Borel–Serre enlargement. The controlled K-theory associ-ated to these compactifications is used to prove the integral Novikov conjecture for arithmetic groups. 1. Statement of the results There is a long history of technique called compactification or attaching a boundary in the study of noncompact symmetric spaces and domains. We will use this term to describe an embedding of the symmetric space as an open subset in a compact Hausdorff space. The boundary points usually carry asymptotic information about the symmetric space which is useful in harmonic analysis and the study of random walks on symmetric spaces. Sometimes these procedures are directly related to compactifications of arithmetic quotients of symmetric spaces. These quotients are moduli spaces of interesting objects, and the boundary points represent the degenerate versions of these objects. A class of constructions which serve both ends is called Satake compactifications. Each Sa-take compactification XS is a union of certain strata attached to the symmetric space X, eac

Year: 2015
OAI identifier: oai:CiteSeerX.psu:10.1.1.515.3741
Provided by: CiteSeerX
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