Renshaw's interests are in algebraic semigroup theory with a particular interest in actions of semigroups and monoids on sets. His early work focused on applications of monoid actions to the theory of semigroup amalgams and on the so-called homological classification theory of monoids. Together with Mojgan Mahmoudi, he initiated the study of flat covers of monoid acts and in particular described strongly flat and condition-(P) covers of cyclic acts.
More recently he started to investigate actions of inverse semigroups and in particular their actions on graphs and trees. The theory of group actions on graphs and trees has proved to be an especially powerful tool and as inverse semigroups are seen more and more to be important in a number of areas of mathematics it is hoped to be able to develop similar tools related to inverse semigroup actions.
Renshaw has also investigated transversals of abundant semigroups and in particular of quasi-adequate semigroups, and has recently studied the discrete log problem and its connections with E-dense semigroups and E-dense actions.