  We study a class of generalisations of Thompson's group $V$ arising naturally
as topological full groups of purely infinite, minimal groupoids. In the
process, we show that the derived subgroup of such a group is 2-generated
whenever it is finitely generated and has no proper characters in full
generality. We characterise this class of groupoids through a number of
group-theoretic conditions on their full groups including vigor, the existence
of suitable embeddings of $V$, and compressibility. We moreover give a complete
abstract characterisation of those groups that arise as either topological full
groups or derived subgroups of purely infinite, minimal groupoids. As an
application, we describe all proper characters of the Brin-Higman-Thompson
groups.
