Brie Finegold

My research is in the areas of geometric group theory and low dimensional topology. As a geometric group theorist, when given a group, I build a topological space on which the group acts by structure-preserving maps. In this way I can explore algebraic properties of the group by understanding the geometry and topology of the space and visa versa. The particular groups I have been investigating for my thesis are special linear groups over commutative rings with one. In order to study these special linear groups, SL(n,R), I constructed a simplicial complex, τn(R), which I call the n th Torus Complex over R, on which SL(n,R) acts simplicially. I have found some relationships between properties of the ring R and properties of τn(R). I also use the machinery of complexes of groups to calculate group presentations for certain special linear groups. Specifically, there is a theorem of Haefliger [4] which describes how, given a group acting cocompactly without inversions on a path connected, simply connected CW complex, one can deduce a presentation. Special linear groups over rings are of interest to many types of mathematicians. Number theorists study quadratic forms via SL(2, R). Topologists and geometric group theorists are interested in arithmetic groups, which are subgroups of SL(n,R). These groups can give rise to hyperbolic manifolds [9] and orbifolds and are also related to lattices. And algebraists who study K-theory often consider presentations for the subgroup of SL(n,R) generated by elementary matrices.