In  we presented a method for constructing different types of examples of finitely generated (f.g.) semigroups having intermediate growth.
Recently we found a family of examples of f.g. semigroups whose growth is intermediate but very close to exponential. Here we discuss the following
Problem. How small or large can the intermediate growth of f.g. inverse semigroups be?
In particular we give the first examples of f.g. inverse semigroups of intermediate growth having minimal (equal to 0) and maximal (equal to 1) Gelfand-Kirillov superdimension.
A part of the talk is a joint work with A. Peluso.
 L. M. Shneerson, Relatively free semigroups of intermediate growth, J. Algebra 235 (2001), 484-546.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.