Scissors Congruence as K-theory

TL;DR

This paper presents a novel approach to scissors congruence groups by constructing them as the zero-level of algebraic K-theory within a Waldhausen category, offering an algebraic perspective.

Contribution

It introduces an alternative construction of scissors congruence groups using algebraic K-theory, connecting geometric concepts with algebraic K-theory frameworks.

Findings

Scissors congruence groups can be realized as K-theory groups.

Provides an algebraic K-theory perspective on geometric scissors congruence.

Establishes a link between group homology and algebraic K-theory.

Abstract

Scissors congruence groups have traditionally been expressed algebraically in terms of group homology. We give an alternate construction of these groups by producing them as the $0$-level in the algebraic $K$-theory of a Waldhausen category.