Generation and simplicity in the airplane rearrangement group (Q6594730)

Given a graph \(\Gamma\) coloured by a set \(\mathrm{Col}\) and a coloured graph \(R_c\) for each colour \(c \in \mathrm{Col}\), called a \emph{replacement graph}, a \emph{full expansion} of \(\Gamma\) is the graph obtained by replacing every edge of \(\Gamma\) with the replacement graph corresponding to its colour. This data forms a \emph{replacement system}. By iteratively applying full expansions, a sequence of graphs is obtained and under certain conditions one can define the limit of this sequence, called the \emph{limit set} of the replacement system, which comes with a natural topology. A \emph{rearrangment} of a limit set is a type of homeomorphism of the limit set, which is combinatorially induced by a graph isomorphism of one of the full expansions.\N\NThe article under review investigates properties of the groups of rearrangements \(T_A\) and \(T_B\) of limit sets of the airplane and basilica replacement systems, which were previously introduced by \textit{J. Belk} and \textit{B. Forrest} [Trans. Am. Math. Soc. 372, No. 7, 4509--4552 (2019; Zbl 1480.20095)]. These limit sets are homeomorphic to the airplane and basilica fractals, which are Julia sets and their rearrangement groups exhibit Thompson group-like behaviour. The author proves that \(T_A\) contains copies of and is generated by Thompson's groups \(F\) and \(T\) and therefore that \(T_A\) is finitely generated. Moreover, the author shows that the commutator subgroup \([T_A,T_A]\) is simple, that \(T_A/[T_A,T_A] \cong \mathbb Z\) and that there are inclusions \(T_B \leqslant T_A \leqslant T\).