 The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.The growth of a finitely generated group measures the asymptotics, as formula_1 of the size of an "n"-ball in the Cayley graph of the group (that is, the number of elements of "G" that can be expressed as words of length at most "n" in the generating set of "G").The study of growth rates of finitely generated groups goes back to the 1950s and is motivated in part by the notion of volume entropy (that is, the growth rate of the volume of balls) in the universal covering space of a compact Riemannian manifold in differential geometry.