Homological stability for symplectic groups via algebraic arc complexes

We use algebraic arc complexes to prove a homological stability result for symplectic groups with slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="two thirds"> <mml:semantics> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {2}{3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for rings with finite unitary stable rank. Symplectic groups are here interpreted as the automorphism groups of <italic>formed spaces with boundary</italic> , 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.