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Indiscrete representations, laminations, and tilings. (English) Zbl 0939.57002
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 225-259 (1999).
The author investigates cocompact representations $$\rho: G\to \text{Isom}(X)$$ of a group $$G$$ in the group of isometries of a proper geodesic metric space $$X$$. However, contrary to the common approach in which $$G$$ acts properly discontinuously so that by the Milnor-Švarc theorem the large scale geometry of $$X$$ is reflected in $$G$$, the author considers indiscrete and/or unfaithful representations. To be more precise, he considers representations $$\rho$$ that can be resolved by a lamination, and calls them laminable. A resolution of $$\rho$$ by a lamination $$\Lambda$$ consists of a properly discontinuous cocompact representation $$\phi$$ of $$G$$ by isometries of a proper quasi-geodesic metric space $$\Lambda$$ with a topological product structure whose leaves are isometrically mapped onto $$X$$ by a $$G$$-equivariant map $$\Lambda\to X$$.
The author gives various characterizations of laminable representations, one among them being expressed in terms of tilings. Many examples are elaborated.
For the entire collection see [Zbl 0910.00040].
##### MSC:
 57M07 Topological methods in group theory 20F65 Geometric group theory 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
##### Keywords:
group representations; laminable representations