Chapter 44 - Forty-plus annotated questions about large topological groups

Abstract

This chapter presents a selection of open problems dealing with large (not locally compact) topological groups. These problems are concerned with extreme amenability (fixed point on compacta property), oscillation stability, universal minimal flows and other aspects of universality, and unitary representations. A topological group G is extremely amenable, or has the fixed point on compacta property, if every continuous action of G on a compact Hausdorff space has a G-fixed point. Some important examples of such groups include: (1) the unitary group U(ℓ2) of the separable Hilbert space ℓ2 with the strong operator topology (that is, the topology of pointwise convergence on ℓ2); (2) the group L1((0, 1),T) of all equivalence classes of Borel maps from the unit interval to the circle with the L1-metric d(f, g) = ∫01|f(x) −g(x)|dx. The chapter presents questions on Levy group, universal minimal flow of the homeomorphism group, and Odell and Schlumprecht's result. Questions on oscillation stable metric space are also discussed.