On higher scissors congruence

TL;DR

This paper extends the classical scissors congruence problem to higher dimensions, solving it for one-dimensional cases and reducing the general problem to group homology computations, revealing the spectrum as a Thom spectrum.

Contribution

It introduces a new approach by identifying the scissors congruence $K$-theory spectrum as a Thom spectrum, linking it to Tits complexes and homotopy orbit spaces.

Findings

Solved the higher Hilbert's Third Problem in one dimension

Reduced higher-dimensional problem to group homology calculations

Identified the spectrum as a Thom spectrum with a Tits complex base

Abstract

We solve the higher version of Hilbert's Third Problem for one-dimensional geometries, and in higher dimensions we reduce the problem to a computation in group homology. Our central result concerns the scissors congruence $K$-theory spectrum of Zakharevich, whose homotopy groups are the correct higher version of the classical scissors congruence groups. We prove that this spectrum is a Thom spectrum, whose base space is the homotopy orbit space of a Tits complex. The relevant computations quickly follow from this more foundational result.