Homological stability for symplectic groups via algebraic arc complexes

TL;DR

This paper proves homological stability for symplectic groups over rings with finite unitary stable rank using algebraic arc complexes, extending stability results to a broader algebraic setting.

Contribution

It introduces a new approach to homological stability for symplectic groups via algebraic arc complexes and boundary formed spaces, with a novel stabilization method.

Findings

Homological stability established for symplectic groups with slope 2/3.

Stability proven for both odd and even symplectic groups.

Uses algebraic arc complexes to analyze automorphism groups of formed spaces.

Abstract

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope 2/3 for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of formed spaces with boundary, which are algebraic analogues of surfaces with boundary, that we also study in the present paper. Our stabilization map is a rank one stabilization in the category of formed spaces with boundary, going through both odd and even symplectic groups.