Associated with any string rewriting system [X ; R] there is a certain
2-complex D (now known as the Squier Complex) on which the free monoid
X* acts on both the left and right. The rewriting system is said to be of
finite derivation type (FDT), respectively,
finite homological type (FHT), if there is a finite set of closed paths in D
such that attaching 2-cells along all translates of these paths yields a
2-complex in which every closed path is null-homotopic,
respectively, null-homologous. So clearly FDT implies FHT, but the converse
has been open. This has been a sticking point to further
development of the geometric theory of rewriting systems. I will discuss
recent joint work with Friedrich Otto showing that the reverse implication
is not true in general.
I will also relate this to recent work of Stuart McGlashan on three-dimensional complexes.
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.