The Basilica Thompson group is not finitely presented

Abstract

We show that the Basilica Thompson group introduced by Belk and Forrest is not finitely presented, and in fact is not of type FP 2 . The proof involves developing techniques for proving non-simple connectedness of certain subcomplexes of CAT(0) cube complexes.

Rearrangement groups of fractals

“…This group is also a special case of a rearrangement group, where the basilica is realized as a limit space of a sequence of graphs in an appropriate way. We conjectured in [1] that T B is not finitely presented, and this was recently proven by S. Witzel and M. Zaremsky using the CATp0q complex we describe here [19].…”

“…T B is generated by four elements. More recently, Witzel and Zaremsky show that T B is not finitely presented [19]. For further discussion, see Example 4.5.…”

Almost-automorphisms of trees, cloning systems and finiteness properties

“…It would also be very interesting to find an example of a finitely generated self-similar (non-contracting, non-F ∞ ) group G ≤ Aut(T d ) such that V d (G) is not of type F ∞ . We have seen examples where G is not even finitely presentable but V d (G) is of type F ∞ (like the Röver group), so in general it seems that any negative finiteness properties would have to come from somewhere other than "carrying over" from G. This question of whether Nekrashevych groups are always F ∞ is especially interesting since they are often simple or virtually simple (see [Nek04, Theorems 9.11 and 9.14] for a precise statement), and to the best of our knowledge it is still an open problem to find simple groups with arbitrary finiteness properties in the world of Thompson-like groups (see also the question in the introduction of [WZ16]). Note that the F 1 -but-not-F 2 case is handled for example by the basilica Thompson group T B [BF15b, WZ16], but even the F 2 -but-not-F 3 case is open.…”

“…There is a large amount of literature devoted to finding finiteness properties of groups in the extended family of Thompson's groups. Usually the groups are of type F ∞ , e.g., see [BM16, Bro87, BFM + 16, FH15, FMWZ13, MPMN16, NSJG14, Thu17], though not always, e.g., Belk-Forrest's basilica Thompson group T B [BF15b] is type F 1 but not F 2 [WZ16]. Also, it is possible to build ad hoc Thompson-like groups with arbitrary finiteness properties, using "cloning systems" [WZ14].…”

Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

“…In Proposition 4.19 we will determine the connectivity properties of I ↑ n (pos). First we need the following useful lemma, which was proved in [WZ16].…”