Controlled algebraic G-theory, II

TL;DR

This paper develops a theory of filtered modules over proper metric spaces with compactification, extending algebraic G-theory control methods to broader classes of groups like CAT(0) groups without finite asymptotic dimension.

Contribution

It generalizes geometric control in algebraic G-theory to filtered modules over proper metric spaces, broadening applicability to CAT(0) groups.

Findings

Developed theory of filtered modules over proper metric spaces

Extended control methods to groups without finite asymptotic dimension

Applicable to CAT(0) groups

Abstract

There are two established ways to introduce geometric control in the category of free modules---the bounded control and the continuous control at infinity. Both types of control can be generalized to arbitrary modules over a noetherian ring and applied to study algebraic $K$-theory of infinite groups. This was accomplished for bounded control in part I of the present paper and the subsequent work of G.~Carlsson and the first author, in the context of spaces of finite asymptotic dimension. This part II develops the theory of filtered modules over a proper metric space with a good compactification. It is applicable in particular to CAT(0) groups which do not necessarily have finite asymptotic dimension.