×

Expanding endomorphisms of crystallographic manifolds. (English) Zbl 0822.20051

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 \(\Gamma\).
Reviewer: H.T.Ku (Amherst)

MSC:

20H15 Other geometric groups, including crystallographic groups
19E99 \(K\)-theory in geometry
19C09 Central extensions and Schur multipliers
57S25 Groups acting on specific manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brown, K. S., Cohomology of Groups (1982), Springer: Springer Berlin · Zbl 0367.18012
[2] Connolly, F.; Kózniewski, T., Rigidity and crystallographic groups I, Invent. Math., 99, 25-48 (1990) · Zbl 0692.57017
[3] Dress, A., Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math., 102, 291-335 (1975) · Zbl 0315.20007
[4] Epstein, D.; Shub, M., Expanding endomorphisms of flat manifolds, Topology, 7, 139-141 (1968) · Zbl 0157.30403
[5] Farkas, D., Crystallographic groups and their mathematics, Rocky Mountain J. Math., 11, 511-551 (1971) · Zbl 0477.20002
[6] Farrell, F. T.; Hsiang, W. C., The Whitehead group of Poly-\((Z\) or finite) groups, J. London Math. Soc., 24, 2, 308-324 (1981) · Zbl 0514.57002
[7] Farrell, F. T.; Hsiang, W. C., Topological characterization of flat and almost flat Riemannian manifolds, Amer. J. Math., 105, 641-672 (1983) · Zbl 0521.57018
[8] Hochschild, G.; Serre, J. P., Cohomology of group extensions, Trans. Amer. Math. Soc., 74, 110-134 (1953) · Zbl 0050.02104
[9] Wolf, J., Spaces of Constant Curvature (1967), McGraw-Hill: McGraw-Hill New York · Zbl 0162.53304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.