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<p>We study connected components of the Morse boundary and their stabilisers. We introduce the notion of <i>point-convergence</i> and show that if the set of non-singleton connected components of the Morse boundary of a finitely generated group <i>G</i> is point-convergent, then every non-singleton connected component is the (relative) Morse boundary of its stabiliser. The above property only depends on the topology of the Morse boundary and hence is invariant under quasi-isometry. This shows that the topology of the Morse boundary not only carries algebraic information but can be used to detect certain subgroups which in some sense are invariant under quasi-isometry.</p>

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Subgroups arising from connected components in the Morse boundary

  • 期刊论文

  • 发表日期：

    2026年5月12日

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  • Annette Karrer,
  • Babak Miraftab,
  • Stefanie Zbinden

摘要

We study connected components of the Morse boundary and their stabilisers. We introduce the notion of point-convergence and show that if the set of non-singleton connected components of the Morse boundary of a finitely generated group G is point-convergent, then every non-singleton connected component is the (relative) Morse boundary of its stabiliser. The above property only depends on the topology of the Morse boundary and hence is invariant under quasi-isometry. This shows that the topology of the Morse boundary not only carries algebraic information but can be used to detect certain subgroups which in some sense are invariant under quasi-isometry.