My current research involves computing the structure of Lie algebras that arise via filtrations of matrix groups by principal congruence subgroups. This Lie algebra structure can be used to obtain information concerning the cohomology of congruence subgroups and related sub-quotients. The paper below collects some results for special linear groups over commutative rings that are free as Z-modules.
I am also interested in questions dealing with linearity of groups, i.e., is there a way to determine if a group is a subgroup of some general linear group? Work of Alex Lubtozky has demonstrated that a group is linear if and only if it admits a filtration satisfying certain finiteness properties. The paper below (with Fred Cohen, Stratos Prassidis, and Marston Condor) relies on Lubotzky's criterion to arrive at conditions under which a semi-direct product of linear groups is linear.
Publications / Preprints
Remarks Concerning Lubotzky's Filtration (with F. Cohen, M. Condor, and S. Prassidis), Pure and Applied Mathematics Quarterly, 8.1 (2012), 79-106.