Type systems and maximal subgroups of Thompson's group $V$

TL;DR

This paper introduces type systems for Thompson's group V, classifies simple ones, and shows their stabilizers are maximal subgroups, revealing new uncountable families of maximal subgroups with unique properties.

Contribution

It defines and classifies simple type systems for V, and demonstrates that their stabilizers form new maximal subgroups, including uncountable families not previously described.

Findings

Finite simple type systems are classified.

Stabilizers of simple type systems are maximal subgroups.

Uncountable families of non-isomorphic maximal subgroups are constructed.

Abstract

We introduce the concept of a type system~$\Part$, 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~$\Stab{V}{\Part}$ 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 non-isomorphic 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.