Non-integrality of some Steinberg modules

Abstract

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ring is Euclidean. We also construct new cohomology classes in the top dimensional cohomology group of the special linear group of some quadratic imaginary number rings.

arXiv comment17 pages. To appear in Journal of Topology

References in corpus (3)

• Stabilization for mapping class groups of 3-manifolds
• Integrality in the Steinberg module and the top-dimensional cohomology of SL_n(O_K)
• The codimension-one cohomology of SL_n Z

Cited by in corpus (7)

• Improved homological stability for certain general linear groups
• On the top dimensional cohomology groups of congruence subgroups of ${\text{SL}}_n(\mathbb{Z})$
• On the codimension-two cohomology of ${\mathrm{SL}}_n(\mathbb{Z})$
• The dualizing module and top-dimensional cohomology group of ${\text{GL}}_n(\mathcal{O})$
• The Steinberg representation is irreducible
• On the generalized Bykovskii presentation of Steinberg modules
• On the top-dimensional cohomology of arithmetic Chevalley groups