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Diameters of homogeneous spaces. (English) Zbl 1029.22031

Motivated by some questions from quantum computing, the authors consider the following question: Given a compact connected Lie group \(G\) with trivial center and a closed subgroup \(H\) of \(G\), how small can the homogeneous space \(G/H\) get without being trivial. They show that if one equips \(G\) with the metric derived from the operator norm of the adjoint representation, then the diameter of \(G/H\) with respect to the induced metric is bounded below by a constant which is roughly 0.12. The proof depends on a result which says that elements of discrete subgroups which are small in norm have to commute.

MSC:

22F30 Homogeneous spaces
53C30 Differential geometry of homogeneous manifolds
22C05 Compact groups
22E46 Semisimple Lie groups and their representations
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