On a substitution subshift related to the Grigorchuk group

The substitution acts on words (finite sequences) over this alphabet as well as on infinite sequences. It is easy to observe that τ has a unique invariant sequence ω that is the limit of words τ (a), k = 1, 2, . . . . Let Ω be the smallest closed set of one-sided infinite sequences over the alphabet {a, b, c, d} that contains ω and is invariant under the shift σ (σ acts on sequences by deleting the first element). The set Ω consists of those sequences for which any finite subword appears somewhere in ω. The restriction of σ to Ω is called a subshift. Since ω is a fixed point of a substitution, this particular subshift is called a substitution subshift. The substitution τ plays an important role in the study of the Grigorhuk group (see the survey [3]). The Grigorchuk group G is a finitely generated infinite group where all elements are of finite order. It has many other remarkable properties as well. The group has four generators a, b, c, d. An important fact is that the substitution τ gives rise to a homomorphism of G to itself. It follows that τ transforms any relator for G into another relator. Although G has no finite presentation, it admits a recursive presentation (due to Lysenok [4]) obtained from a finite set of relators by repeatedly applying τ :