Type systems and maximal subgroups of Thompson's group V

We introduce the concept of a type system 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 Stab_V(𝒫) 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.