Expanding endomorphisms of crystallographic manifolds (Q1341409)

Let \(\Gamma\) be a crystallographic group with associated exact sequence \(0 \to A \to \Gamma \to G \to 1\) and let \(M_ \Gamma\) be the flat crystallographic manifold (i.e., the \(G\)-equivariant torus \(R^ n /A\)) associated to \(\Gamma\). The author constructs a new crystallographic group, a quotient of \(\Gamma/ A_ C\), where \(A_ C\) is the sum of all 1-dimensional \(G\)-submodules of \(A\). If \(A_ C\) is trivial, this case was studied by Epstein and Shub (1968). If \(A_ C \neq 0\), the author shows that the endomorphisms of \(M_ \Gamma\) expand in certain directions transverse to the fibres of the map \(M_ \Gamma \to M_ \Delta\). The existence of such expanding maps is of interest to the study of the \(K\)-theory as well as the controlled \(K\)-theory of \(\Gamma\).