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

Jonathan Lopez

• 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.Z Z/p).Fp tFp[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.