Geodesic
Finitely generated subgroups of branch groups and subdirect products of just infinite groups
Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1128-1145

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The aim of this paper is to describe the structure of finitely generated subgroups of a family of branch groups containing the first Grigorchuk group and the Gupta–Sidki $3$-group. We then use this to show that all the groups in this family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.
Keywords: just infinite groups, subdirect products, branch groups. 
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     author = {R. I. Grigorchuk and P.-H. Leemann and T. V. Nagnibeda},
     title = {Finitely generated subgroups of branch groups and subdirect products of just infinite groups},
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     pages = {1128--1145},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_6_a3/}
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R. I. Grigorchuk; P.-H. Leemann; T. V. Nagnibeda. Finitely generated subgroups of branch groups and subdirect products of just infinite groups. Izvestiya. Mathematics , Tome 85 (2021) no. 6, pp. 1128-1145. http://geodesic.mathdoc.fr/item/IM2_2021_85_6_a3/