Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this talk I will discuss how a number of easily-defined expansions on finite automata lead to simplifications of the corresponding semigroups. In particular that every finite semigroup can be finitely expanded so that the right Cayley graph of the expansion closely resembles the right Cayley graph of a free Burnside semigroup.