Michael Megrelishvili

Key subgroups in the Polish group of all automorphisms of the rational circle.

Extending some results of a joint work with E. Glasner (2021), we continue to study the Polish group G:=\mathrm{Aut} (\mathbb{Q}_0) of all circular order preserving permutations of \mathbb{Q}_0 with the pointwise topology, where \mathbb{Q}_0=\mathbb{Q}/\mathbb{Z} is the rational discrete circle. We show that certain subgroups H of G:=\mathrm{Aut} (\mathbb{Q}_0) are inj-key (i.e., H distinguishes weaker Hausdorff group topologies on G) but not co-minimal in G. This counterexample answers a question from a recent joint work with M. Shlossberg (2025) and is inspired by a question proposed (during the conference: Algebra, Topology and Their Interactions 2024 ) by V. Pestov about Polish groups G with metrizable universal minimal G-flow M(G). It is an open problem to study Pestov’s question in its full generality. We are also going to discuss some additional open questions.

The following (algebraic, topological and dynamical) concepts are important in this project:

  1. Key subgroups (as a new minimality condition in topological groups).
  2. Circularly ordered dynamical G-systems.
  3. Universal minimal dynamical G-systems M(G).
  4. Maximal G-compactifications of G-spaces.

References

[1] E. Glasner and M. Megrelishvili, Circular orders, ultrahomogeneity and topological groups, AMS Contemporary Math. book series 772 “Topology, Geometry, and Dynamics: Rokhlin-100” (ed.: A.M. Vershik, V.M. Buchstaber, A.V. Malyutin) 2021, pp. 133—154. ArXiv:1803.06583.

[2] M. Megrelishvili and M. Shlossberg, Key subgroups in topological groups, Forum Math. 2025, doi.org/10.1515/forum-2023-0418. ArXiv:2309.06785.

[3] M. Megrelishvili, Key subgroups in the Polish group of all automorphisms of the rational circle. ArXiv:2410.17905.