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A survey on divisible convex sets. (English) Zbl 1154.22016

Ji, Lizhen (ed.) et al., Geometry, analysis and topology of discrete groups. Selected papers of the conference on geometry, topology and analysis of locally symmetric spaces and discrete groups, Beijing, China, July 17–August 4, 2006. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-126-1/hbk). Advanced Lectures in Mathematics (ALM) 6, 1-18 (2008).
Since J. P. Benzécri’s paper [Bull. Soc. Math. Fr. 88, 229–332 (1960; Zbl 0098.35204)] the understanding of what is now called divisible convex sets has advanced considerably. The main objects of study are properly convex open subsets \(\Omega\) of \(n\)-dimensional real projective space for which there exists a discrete group \(\Gamma\) of projective transformations preserving \(\Omega\) such that \(\Gamma\backslash \Omega\) is compact.
In this paper the author, who has a longstanding interest in this area and has contributed significantly to it, gives a brief survey, without proofs, of the major themes and developments in divisible convex sets and clearly conveys the flavour of the area and its many connections to other areas in mathematics. The section headings of the paper are symmetric convex sets, first examples, strict convexity and regularity of \(\Omega\), irreducibility of \(\Omega\) and of \(\Gamma\), moduli spaces of representations, parametrizations in dimension 2, more examples, the 3-dimensional case, and a comprehensive list of references.
For the entire collection see [Zbl 1144.22001].

MSC:

22E40 Discrete subgroups of Lie groups
20H15 Other geometric groups, including crystallographic groups
57N10 Topology of general \(3\)-manifolds (MSC2010)
57S30 Discontinuous groups of transformations

Citations:

Zbl 0098.35204
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