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

Let R be a commutative ring that is free of rank k as an abelian group, p a prime, and SLn(R) the special linear group. We show that the Lie algebra associated to the filtration of SLn(R) by p-congruence subgroups is isomorphic to the tensor product sln(R⊗ZZ/p)⊗FptFp[t], the Lie algebra of polynomials with zero constant term and coefficients n×n traceless matrices with entries polynomials in k variables over Fp.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≥3. We show that the cohomology groups of the level pr-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 pr-congruence subgroup has dimension at least two as an Fp-vector space.