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Abstract We show that R. Thompson’s group Tis a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements ofVas products of transpositions of basic clopen sets in the Cantor space $$\mathfrak {C}$$ $C$. 

Abstract We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups. 

Abstract Given$$g \in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$, let$$\Sigma _g$$ ${\Sigma }_g$denote the closed surface of genusgwith a Cantor set removed, if$$g<\infty $$ $g<\infty$; or the blooming Cantor tree, when$$g= \infty $$ $g=\infty$. We construct a family$$\mathfrak B(H)$$ $B\left(H\right)$of subgroups of$${{\,\textrm{Map}\,}}(\Sigma _g)$$ $\text{Map}\left({\Sigma }_g\right)$whose elements preserve ablock decompositionof$$\Sigma _g$$ ${\Sigma }_g$, andeventually like actlike an element ofH, whereHis a prescribed subgroup of the mapping class group of the block. The group$$\mathfrak B(H)$$ $B\left(H\right)$surjects onto an appropriate symmetric Thompson group of Farley–Hughes; in particular, it answers positively. Our main result asserts that$$\mathfrak B(H)$$ $B\left(H\right)$is of type$$F_n$$ $F_n$if and only ifHis. As a consequence, for every$$g\in \mathbb N \cup \{0, \infty \}$$ $g\in N\cup \{0,\infty \}$and every$$n\ge 1$$ $n\ge 1$, we construct a subgroup$$G <{{\,\textrm{Map}\,}}(\Sigma _g)$$ $G<\text{Map}\left({\Sigma }_g\right)$that is of type$$F_n$$ $F_n$but not of type$$F_{n+1}$$ $F_{n+1}$, and which contains the mapping class group of every compact surface of genus$$\le g$$ $\le g$and with non-empty boundary. 

Abstract We introduce “braided” versions of self-similar groups and Röver–Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the self-similar groups are what we call “self-identical.” In particular, we use a braided version of the Grigorchuk group to construct a new group called the “braided Röver group,” which we prove is of type$$\operatorname {\mathrm {F}}_\infty $$. Our techniques involve using so-calledd-ary cloning systems to construct the groups, and analyzing certain complexes of embedded disks in a surface to understand their finiteness properties. 

We introduce the concept of a type system  $\mathcal{P}$, that is, a partition on the set of finite words over the alphabet  $\{0,1\}$compatible with the partial action of Thompson’s group  $V$, and associate a subgroup  ${\mathrm{Stab}}_V<\#comment/>\left(\mathcal{P}\right)$of  $V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of  $V$. We also find an uncountable family of pairwise nonisomorphic maximal subgroups of  $V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$(both related to primitivity) are each satisfied only by $V$itself, giving new ways to recognise when a subgroup of $V$is not actually proper. 

We study the finiteness properties of the braided Higman–Thompson groupbV_{d,r}(H)with labels inH\leq B_d, andbF_{d,r}(H)andbT_{d,r}(H)with labels inH\leq PB_d, whereB_dis the braid group withdstrings andPB_dis its pure braid subgroup. We show that for alld\geq 2andr\geq 1, the groupbV_{d,r}(H)(resp.bT_{d,r}(H)orbF_{d,r}(H)) is of typeF_nif and only ifHis. Our result in particular confirms a recent conjecture of Aroca and Cumplido.