The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

TL;DR

This paper investigates a coarse version of the Farrell--Jones conjecture, proving it for box spaces of residually finite groups under certain conditions using controlled category theory.

Contribution

It introduces a coarse isomorphism conjecture and proves it for box spaces of residually finite groups with an isomorphic Farrell--Jones assembly map.

Findings

Proves the coarse isomorphism conjecture for specific box spaces

Uses controlled category theory to relate conjectures

Establishes conditions under which the conjecture holds

Abstract

In this article we study a coarse version of the $K$-theoretic Farrell--Jones conjecture we call coarse or bounded isomorphism conjecture. Using controlled category theory we are able to translate this conjecture for asymptotically faithful covers into a more familiar form. This allows us to prove the conjecture for box spaces of residually finite groups whose Farrell--Jones assembly map with coefficients is an isomorphism.