Regular Algebraic $K$-theory for groups -- Part I

TL;DR

This paper introduces a new homology theory called regular algebraic K-theory for groups, which differs from traditional group homology and extends to rings via a functorial approach involving infinite-dimensional general linear groups.

Contribution

It develops a novel algebraic K-theory framework for groups and rings, connecting homology with algebraic structures through a functorial construction.

Findings

Defines regular algebraic K-theory as a homology theory for groups.

Establishes a functorial method to extend K-theory to rings.

Links the theory to the commutator subgroup of infinite-dimensional general linear groups.

Abstract

Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected (but different from) group homology. It also gives a version of algebraic $K$-theory for rings by the simple functorial mapping assigning to a ring $R$ the (perfect>) commutator subgroup $E ( R )$ of the infinitedimensional general linear group over $R$.