Maps related to Grigorchuk’s group

Grigorchuk’s group of intermediate growth can be represented, through its action on the infinite binary rooted tree, as the automorphism group of a regular map GG on a non-compact surface. A theory of growth of maps is developed, and it is shown that GG has intermediate growth. Some compact and non-compact quotients of GG are described, and it is shown how these ideas may be extended to the generalised Grigorchuk groups.