The NIP Hausdorffness conjecture for Ellis groups

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Let MM be a first-order structure with NIP. Let GG be a group definable in MM, let N≻MN\succ M be ∣N∣+|N|^+-saturated, and let uMu\mathcal{M} be the Ellis group of the flow (G,SG,M(N))(G,S_{G,M}(N)), equipped with its τ\tau-topology.

NIP Hausdorffness conjecture. If MM has NIP, then uMu\mathcal{M} is Hausdorff.

In general, non-Hausdorff Ellis groups are known, but the source reports no such example with NIP. Thus the assertion remains open in the stated generality.

References

Primary source

Krzysztof Krupiński and Anand Pillay, “Generalized locally compact models for approximate groups”, arXiv:2310.20683 (2026).

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