Automorphisms of the Discrete Heisenberg Group

This note describes some computational results that were obtained while studying the associated group of automorphisms Aut(H), [2]. Specifically, we give an explicit description of a splitting of this group as an extension of GL(2,Z) by Z. That such a splitting exists could, in principle, be shown by computing a 2-cocycle for the extension and showing that it represents 0 in the cohomology group H(GL(2,Z),Z) ≈ Z2, where, here, Z has the standard (left)-GL(2,Z)-module structure. However, this computation is no easier than the direct construction—indeed, it may be substantially harder— and does not appear to be as useful. We shall also include a proof of the isomorphism H(GL(2,Z),Z) ≈ Z2, which is presumably well known, but the main point of this note is to provide explicit computations in Aut(H), such as that of the mentioned splitting that may be of use to someone working with this automorphism group. A useful reference for the cohomology computations is [1].