On Lie algebras and cohomology associated to congruence subgroups

Let R be a commutative ring that is free of rank k as an abelian group, V a Z -basis for R, p a prime, and Gn(R) an n x n matrix group. We show that the Lie algebra associated to the filtration of Gn(R) by p-congruence subgroups is isomorphic to the tensor product g⊗Fp tFp t , the Lie algebra of polynomials with zero constant term and coefficients in g , where g is a restricted Lie algebra over Fp that depends on Gn(R). In this paper, we consider the cases Gn( R) = SL(n, R), Gn (R) = PSL(n, R), Gn(R) = GL(n, R), and Gn(R) = Sp(4, R). For these cases, we have g=sln &parl0;Fp V&parr0;, g=sln&parl0;F pV &parr0;,g=gln&parl0; FpV &parr0; , and g=sp4 &parl0;Fp V&parr0; , respectively.;In many cases, we are able to use the underlying group structure to explicitly compute the structure of the Lie algebra associated to the filtration of Gn(R) by p-congruence subgroups. In particular, we are able to show that successive filtration quotients are isomorphic to elementary abelian p-groups. We also obtain some homological results concerning these Lie algebras and the cohomology of congruence subgroups.;We use the Lie algebra structure along with the Lyndon-Hochschild-Serre spectral sequence to compute the d2 homology differential for certain central extensions involving quotients of p-congruence subgroups. We also use the underlying group structure to obtain several homological results by exhibiting various subgroups of the level pr-congruence subgroup. For example, we compute the first homology group of the level p-congruence subgroup for Gn(R) = SL( n, R) and n ≥ 3. We show that the cohomology groups of the level pr-congruence subgroup are not finitely generated for Gn( R) = SL(n, Z [t]) and r ≥ 1. Finally, we show that for Gn(R) = SL( n, Z [i]) and r ≥ 1, where Z [i] denotes the Gaussian integers, the second cohomology group of the level pr-congruence subgroup has dimension at least two as an Fp -vector space.