2010/01/13 by Jonathan Lopez, Lopez, Jonathan
Mathematics · #17B45 #20H05 #20J06 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.AT #math.GR #math.NT #msc:17B45 #msc:20H05 #msc:20J06
paper · pdf · doi:10.48550/arxiv.1001.2071
22 pages
openalex publication_date 2010/01/13 · arxiv created 2012/09/06 · arxiv updated 2012/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfraksln(R⊗\Z\Z/p)⊗\Fpt\Fp[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 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. 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.