Forty Annotated Questions about Large Topological Groups

This is a selection of open problems dealing with “large” (non locally compact) topological groups and concerning 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 Gfixed point. Here are some important examples of such groups. Example 1. The unitary group U(l) of the separable Hilbert space l with the strong operator topology (that is, the topology of pointwise convergence on l) (Gromov and Milman [28]). Example 2. The group L((0, 1),T) of all equivalence classes of Borel maps from the unit interval to the circle with the L-metric d(f, g) = ∫ 1 0 |f(x) − g(x)| dx (Glasner [22], Furstenberg and Weiss, unpublished). Example 3. The group Aut (Q,≤) of all order-preserving bijections of the rationals, equipped with the natural Polish group topology of pointwise convergence on Q considered as a discrete space and, as an immediate corollary, the group Homeo+[0, 1] of all homeomorphisms of the closed unit interval, preserving the endpoints, equipped with the compact-open topology (the present author [39]). The above property is not uncommon among concrete “large” topological groups coming from diverse parts of mathematics. In addition to the above quoted articles, we recommend [21, 34] and the book [42]. The group in example 2 is monothetic, that is, contains a dense subgroup isomorphic to the additive group of integers Z. Notice that every abelian extremely amenable group G is minimally almost periodic, that is, admits no non-trivial continuous characters (the book [13] is a useful reference): indeed, if χ : G → T is such a character, then (g, z) 7→ χ(g)z defines a continuous action of G on T without fixed points. The converse remains open. Question 1 (Eli Glasner [22]). Does there exist a monothetic topological group 1001 ? that is minimally almost periodic but not extremely amenable? 2000 Mathematics Subject Classification. Primary: 22A05 Secondary: 43A05, 43A07, 54H15. Research program by the author has been supported by the NSERC operating grant (2003–07) and the University of Ottawa internal grants (2002-04 and 2004-08).