Locally symmetric spaces and K-theory of number fields

TL;DR

This paper explores the relationship between locally symmetric spaces and algebraic K-theory of number fields, introducing invariants derived from fundamental classes and extending constructions to cusped spaces.

Contribution

It introduces a new invariant in algebraic K-theory associated with locally symmetric spaces and generalizes the construction to spaces with cusps of R-rank one.

Findings

The invariant is nontrivial in certain cases.

Extension of the construction to cusped spaces of R-rank one.

Provides new insights into the K-theory of number fields.

Abstract

For a closed locally symmetric space M=\Gamma\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of R-rank one.