Maximal subgroups of topological full groups of shifts of finite type

Abstract

We study maximal subgroups of topological full groups of irreducible one-sided shifts of finite type whose associated groupoids have Cantor unit space. For each wide open subgroupoid of such a groupoid, we construct a subgroupoid graph and identify the subgroupoid with a full compact open reduction of the corresponding countable graph groupoid. Every minimal wide open subgroupoid is either AF or purely infinite. A proper wide open subgroupoid is prime if it admits no proper intermediate wide open subgroupoid. We prove a trichotomy for prime subgroupoids: they are minimal, arise from a clopen bipartition, or have a unique nonempty proper closed invariant set, which is nowhere dense. For a prime subgroupoid, the normalizer of its topological full group is maximal in the ambient topological full group precisely when it maps onto the ambient abelianization. An analogous criterion holds in every subgroup containing the ambient commutator subgroup. For a minimal prime subgroupoid, its topological full group is maximal precisely when it equals its normalizer and the inclusion induces a surjection on first groupoid homology. We also describe all fully amplified maximal subgroups, namely those closed under local agreement. For Thompson's group $V$, the stabilizer of every simple type system is maximal, extending results of Belk, Bleak, Quick and Skipper for finite type systems and certain classes of infinite ones. Finally, every subgroup containing the ambient commutator subgroup has continuum many pairwise nonisomorphic maximal subgroups acting minimally on the unit space.