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arXiv · 1807.08009

On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki $3$ group

Rachel Skipper·Phillip Wesolek
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Abstract

We study the Cantor--Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for $G$ either the Grigorchuk group or the Gupta--Sidki $3$ group, the Cantor--Bendixson rank of $\mathrm{Sub}(G)$ is $\omega$. For each natural number $n$, we additionally characterize the subgroups of rank $n$ and give a description of subgroups in the perfect kernel.

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Rachel Skipper, Phillip Wesolek. 2018-07-20. On the Cantor-Bendixson rank of the Grigorchuk group and the Gupta-Sidki $3$ group. https://arxiv.org/abs/1807.08009

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