On passage to over-groups of finite indices of the Farrell-Jones conjecture

TL;DR

This paper investigates whether the Farrell-Jones conjecture remains valid when passing to over-groups of finite index, using controlled algebra and an extended induction theorem in algebraic K- and L-theories.

Contribution

It extends the classical induction theorem to twisted coefficients in additive categories and applies it to analyze the finite index passage problem in the Farrell-Jones conjecture.

Findings

Extended induction theorem for K- and L-theories with twisted coefficients.

Reductions for the finite index problem in the Farrell-Jones conjecture.

Detailed proof of the extended induction theorem.

Abstract

We use the controlled algebra approach to study the problem that whether the Farrell-Jones conjecture is closed under passage to over-groups of finite indices. Our study shows that this problem is closely related to a general problem in algebraic $K$- and $L$-theories. We use induction theory to study this general problem. This requires an extension of the classical induction theorem for $K$- and $L$- theories of finite groups with coefficients in rings to with twisted coefficients in additive categories. This extension is well-known to experts, but a detailed proof does not exist in the literature. We carry out a detailed proof. This extended induction theorem enables us to make some reductions for the general problem, and therefore for the finite index problem of the Farrell-Jones conjecture.