arXiv subjects

Matthew Scalamandre

Publications and source records attributed to Matthew Scalamandre.

2 recordsLinked to original sources

arXiv2026

High-degree cohomology of congruence subgroups of $\text{SL}_n(\mathcal{O})$ via cohomology of $S$-arithmetic groups

If $\mathfrak{p}$ is a prime ideal of a number ring $\mathcal{O}$, then the top-degree cohomology of the principal congruence subgroup of level $\mathfrak{p}$ is naturally a representation of $\text{SL}_n(\mathcal{O}/\mathfrak{p}).$ We prove that the multiplicity of the Steinberg representation in this cohomology space is one. When $\mathcal{O}$ is Euclidean and $\mathfrak{p}$ is suitably small -- for example a universal side divisor -- then we prove that the multiplicity of the Steinberg representation in the next-highest-degree cohomology space is zero. Our proof relies on a computation of the cohomology of an $S$-arithmetic group ouside of a linear range of degrees, derived from work of Blasius--Franke--Grunewald.

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arXiv2023

A Solomon-Tits theorem for rings

An analog of the Tits building is defined and studied for commutative rings. We prove a Solomon-Tits theorem when $R$ either satisfies a stable range condition, or is the ring of $S$-integers of a global field. We then define an analog of the Steinberg module of $R$, and study it both as a $\mathbb{Z}$-module and as a representation. We find the rank of Steinberg when $R$ is a finite ring, and compute the length of $\text{St}_2(R)\otimes\mathbb{Q}$ as a $\text{GL}_2(R)$-representation when $R$ is uniserial. As an application of these results, we produce a lower bound for the rank of the top-dimensional cohomology of principal congruence subgroups of nonprime level.

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