A self-similar action (G,X) consists of a group G and finite alphabet X along with an action of the group on the free monoid X* consisting of all finite words in the letters of X. Mark Lawson showed that a self-similar action could be realised as a Zappa-Szép product semigroup X* x G. A universal Toeplitz algebra can be constructed for a self-similar action with a Cuntz-Pimsner algebra quotient. We show that

a generalised self-similar Toeplitz algebra can be constructed for a Zappa-Szép product with a boundary quotient generalising the Cuntz-Pimsner algebra of a self-similar action. We use this to describe new presentations of C*-algebras associated to semigroups constructed by Xin Li. Some examples include the Baumslag-Solitar groups and the affine semigroup over the natural numbers.

This is joint work with Nathan Brownlowe, Jacqui Ramagge, and Dave Robertson.