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Type systems and maximal subgroups of Thompson’s group 𝑉

https://doi.org/10.1090/btran/204

Belk, James; Bleak, Collin; Quick, Martyn; Skipper, Rachel (April 2025, Transactions of the American Mathematical Society, Series B)

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} <#comment/> (P)

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.

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