Relative completions of linear groups over Z[t] and Z[t,t^{-1}]

TL;DR

This paper computes the relative completions of certain linear groups over polynomial and Laurent polynomial rings, extending classical results and providing partial insights into their rational second cohomology.

Contribution

It generalizes the classical Malcev completion to groups over Z[t] and Z[t,t^{-1}], offering new computational methods and partial cohomology results.

Findings

Computed the relative completions of SL_n(Z[t]) and SL_n(Z[t,t^{-1}])

Extended Malcev completion concepts to polynomial rings

Provided partial calculations of the groups' rational second cohomology

Abstract

We compute the completion of the groups SL_n(Z[t]) and SL_n(Z[t,t^{-1}]) relative to the obvious homomorphisms to SL_n(Q); this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.