The Torus Complex and Special Linear Groups over Rings.

摘要

Let R be a commutative ring with one. We produce a canonical means for associating to R a family of simplicial complexes {taun(R)| n ∈ N } which we call torus complexes on which SL( n, R) acts simplicially. We show that for n > 2 and R is a Principal Ideal Domain, the nth torus complex over R is connected with diameter two. Also, if SL(2, R) is generated by transvections, then tau2(R) is connected. We show that the 3rd torus complex over a Euclidean ring R is simply connected, and we use the theory of complexes of groups to derive presentations for SL(3, R).;One question that this approach raises is whether a ring can be characterized by the topological properties of its family of torus complexes. Also, the method by which the torus complex was constructed is analogous to the curve complex construction (it comes out of a topological definition), and suggests a version for algebraic varieties might be of interest.;Also, there is a recent interest in the finite presentability of certain special linear groups over polynomial rings which this approach may be useful for exploring.