Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures

Vivian He; Davide Spriano; Stefanie Zbinden

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arXiv · 2407.18863

Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures

Vivian He·Davide Spriano·Stefanie Zbinden

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Abstract

We show that the Morse boundary of a Morse local-to-global group is $\sigma$-compact. Moreover, we show that the converse holds for small cancellation groups. As an application, we show that the Morse boundary of a non-hyperbolic, Morse local-to-global group that has contraction does not admit a non-trivial stationary measure. In fact, we show that any stationary measure on a geodesic boundary of such a groups needs to assign measure zero to the Morse boundary. Unlike previous results, we do not need any assumptions on the stationary measures considered.

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BibTeX RIS

Vivian He, Davide Spriano, Stefanie Zbinden. 2024-07-26. Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures. https://arxiv.org/abs/2407.18863

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