The author studies existence and properties of abelian subgroups in (bi- )automatic groups and in groups which act properly and cocompactly on simply-connected spaces of nonpositive curvature.
The main technical result is the following: let \(\mathcal A\) be a finite set and \(\mu: {\mathcal A}^* \to \Gamma\) a monoid epimorphism to a group \(\Gamma\). Suppose that \(H \subset \Gamma\) is an abelian subgroup for which there exists a regular language \({\mathcal L} \subset {\mathcal A}^*\) such that \(\mu({\mathcal L}) = H\). Then, there exist only finitely many commensurability classes of subgroups \(K'\) in \(H\) such that each class is represented by a subgroup \(K \in K'\), \(\mu^{-1}(K) \cap {\mathcal L}\) is a regular language. As an application, the author gives a criterion (in terms of regular languages) for a finitely generated group to contain a subgroup isomorphic to \(Z \times Z\). Another result is the following: Let \(X\) be a simply-connected geodesic metric space of nonpositive curvature and \(\Gamma\) a finitely generated group which acts properly and cocompactly by isometries on \(X\). Then the author gives a sufficient condition for every abelian subgroup of \(\Gamma\) to be virtually cyclic in terms of the combing associated to the action.