Metric relations in groups of tree automorphisms

Abstract

For groups of tree automorphisms, we couch an analog for a dimension that generalizes the ordinary dimension to the case of groups of direct power. On a Grigorchuk 2-group, that generalized dimension has the additivity property. Generalized dimensions of centralizers of elements in the Grigorchuk group are computed. The dual notion of a codimension is introduced. The relationship between the codimension of a subgroup and the dimension of this subgroup's centralizer is explicated.