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We show that the Basilica Thompson group introduced by Belk and Forrest is not finitely presented, and in fact is not of type FP. The proof involves developing techniques for proving non-simple connectedness of certain subcomplexes of CAT(0) cube complexes.
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Stefan Witzel, Matthew C.B. Zaremsky, The Basilica Thompson group is not finitely presented. Groups Geom. Dyn. 13 (2019), no. 4, pp. 1255–1270