On the algebraic K-theory of 3-manifold groups

TL;DR

This paper describes the algebraic K-theory and Whitehead groups of 3-manifold groups using geometric decompositions, the Farrell-Jones conjecture, and Nil-groups, providing a comprehensive algebraic understanding of these fundamental groups.

Contribution

It offers new descriptions of algebraic K-theory groups of 3-manifold groups based on their geometric and subgroup structures, utilizing advanced conjectures and decompositions.

Findings

Whitehead groups expressed via finite subgroups and Nil-groups

Application of Farrell-Jones isomorphism conjecture to 3-manifold groups

Use of prime and JSJ-decompositions with geometrization theorem

Abstract

We provide descriptions of the Whitehead groups, and the algebraic $K$-theory groups, of the fundamental group of a connected, oriented, closed $3$-manifold in terms of Whitehead groups of their finite subgroups and certain Nil-groups. The main tools we use are: the K-theoretic Farrell-Jones isomorphism conjecture, the construction of models for the universal space for the family of virtually cyclic subgroups in 3-manifold groups, and both the prime and JSJ-decompositions together with the well-known geometrization theorem.