Lie algebras and cohomology of congruence subgroups for SL_n(R)

TL;DR

This paper investigates the Lie algebra structure of p-congruence subgroups of SL(n,R), computes their homology and cohomology groups, and reveals non-finite generation and dimension properties for specific cases.

Contribution

It introduces an explicit Lie algebra model for p-congruence subgroups of SL(n,R) and computes their homological and cohomological properties, including non-finite generation results.

Findings

Lie algebra of p-congruence subgroups is isomorphic to a tensor product involving polynomial matrices.

Computed the first homology group of level p-congruence subgroups for n≥3.

Showed cohomology groups are not finitely generated for n=2 and R=Z[t].

Abstract

Let $R$ be a commutative ring that is free of rank $k$ as an abelian group, $p$ a prime, and $SL(n,R)$ the special linear group. We show that the Lie algebra associated to the filtration of $SL(n,R)$ by $p$-congruence subgroups is isomorphic to the tensor product $\mathfrak{sl}_n(R\otimes_{\Z}\Z/p)\otimes_{\F_p}t\F_p[t]$, the Lie algebra of polynomials with zero constant term and coefficients $n\times n$ traceless matrices with entries polynomials in $k$ variables over $\F_p$. We use the Lie algebra structure along with the Lyndon-Hochschild-Serre spectral sequence to compute the $d^2$ homology differential for certain central extensions involving quotients of $p$-congruence subgroups. We also use the underlying group structure to obtain several homological results. For example, we compute the first homology group of the level $p$-congruence subgroup for $n\geq3$. We show that the cohomology groups of the level $p^r$-congruence subgroup are not finitely generated for $n=2$ and $R=\Z[t]$. Finally, we show that for $n=2$ and $R=\Z[i]$, the Gaussian integers, the second cohomology group of the level $p^r$-congruence subgroup has dimension at least two as an $\F_p$-vector space.