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Divisible convex sets. III. (Convexes divisibles. III). (French) Zbl 1085.22006
[Parts I and II by the same author in S. G. Dani (ed.) et al., Algebraic groups and arithmetic. New Delhi 2004, 339–374 (2004; Zbl 1084.37026); Duke Math. J. 120, 97–120 (2003; Zbl 1037.22022)].
Let \(\Delta_0\) be a group of finite type and \({\mathcal F}_{\Delta_0}\subset \text{Hom}(\Delta_0,\text{PGL}(\mathbb{R}^m))\) be the subset of faithful representations for which there exists a properly convex \(\Delta_0\)-invariant open subset \(\Omega\) in \(\mathbb P(\mathbb{R}^m)\) such that the quotient \(\Delta_0\backslash \Omega\) is compact. Koszul proved that \({\mathcal F}_{\Delta_0}\) is open in \(\text{ Hom}(\Delta_0,\text{PGL}(\mathbb R^m))\) which can be viewed as a closed subset of a finite product of copies of \(\text{PGL}(\mathbb{R}^m)\). The author studies the question under which conditions \({\mathcal F}_{\Delta_0}\) is also closed. It turns out that this is the case if and only if the virtual center of \(\Delta_0\) is trivial. Another equivalent condition is the existence of a strongly irreducible representation in \({\mathcal F}_{\Delta_0}\). The study of linear groups preserving a convex cone plays a key role in the proof.

MSC:
22E40 Discrete subgroups of Lie groups
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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