Twisted homology stability of O_n for valuation rings

TL;DR

This paper proves homology stability for orthogonal groups over valuation rings under certain conditions, extending previous work and applying to fields beyond the reals.

Contribution

It extends Vogtmann's argument to valuation rings, establishing homology stability for $O_n(A)$ with twisted coefficients under new arithmetic conditions.

Findings

Homology stability holds for $O_n(A)$ over henselian valuation rings with finite Pythagoras number residue fields.

Results include extensions of Vogtmann's work and apply to non-real fields.

Provides analogues for fields other than $\,\mathbb{R}$ in scissor congruence studies.

Abstract

In this article, we extend an argument of Vogtmann in order to show homology stability of the Euclidean orthogonal group $O_n(A)$ when $A$ is a valuation ring subject to arithmetic conditions on either its residue or its quotient field. In particular, it is shown that if $A$ is a henselian valuation ring, then the groups $O_n(A)$ exhibit homology stability if the residue field of $A$ has finite Pythagoras number. Our results include those of Vogtmann, and hold with various twisted coefficients. Using these results, we give analogues for fields $F\neq\mathbb R$ of some computations that appear in the study of scissor congruences.