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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)
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