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

James Belk, Collin Bleak, Martyn Quick, Rachel Skipper

Published 2022-06-25Version 1

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~$V$. In particular, we show that there are uncountably many maximal subgroups of~$V$ that occur as the stabilizers of simple type systems and do not arise in the form of previously known maximal subgroups. Finally, we consider two conditions for subgroups of~$V$ that could be viewed as related to the concept of primitivity. We show that in fact only $V$~itself satisfies either of these conditions.