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Revised Newelski’s conjecture (Hausdorff τ‑topology on ideal groups)

Show that for any NIP structure M and any definable group G, the τ‑topology on the ideal group uM of the G(M)‑flow S_{G,M}(U) is Hausdorff.

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Background

The τ‑topology is a central tool in topological dynamics. Newelski’s original Ellis group conjecture failed in full generality, motivating a revised version asking only that the ideal group be Hausdorff in the τ‑topology.

The authors prove this conjecture for countable NIP groups by leveraging Glasner’s structure theorem for tame metrizable flows. The general (uncountable) case remains open.

References

\begin{conjecture}\label{conjecture: revised Newelski's conjecture}Conjecture 5.3 If $M$ is NIP, then the $\tau$-topology on $uM$ is Hausdorff. \end{conjecture}

— Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture (2406.00912 - Chernikov et al., 3 Jun 2024) in Section 5 (Topological dynamics); subsection ‘Revised Newelski’s conjecture’