Growth of self-similar groups
by
Yuriy Leonov
Odessa National Academy of Telecommunications

Let T_d is d-regular rooted tree and G £ Aut T_d. Let u be a vertex of T_d and g Î G. Let us denote by y_u(g) the restriction of the action of the element g on the tree T_d to the subtree T(u) with root u. Let us define y_u(G)={y_u(g); g Î G}. A group G is called self-similar if for any vertex u y_u(G)=G after the identification of the tree T_d with a subtree T(u). We denote T_d^(k)={u Î T_d ; dist(u)=k}, where dist(u) is a distance between u and the root vertex of the tree T_d. We say that a finitely generated self-similar group G is ordinary if there is system of generators S and a natural k such that

lS(g) £ å u Î Td(k)  lS(yu(g)),

where l(*) is the length of an element * with respect to the set of generators S. For an ordinary group G and for a natural number k we consider the set without contracting:

FG, S, k(n) = {g Î G | lS(g) = å u Î Td(k)  lS(yu(g)) £ n}.

Denote f_{G, S, k}(n)=|F_{G, S, k}(n)|. A function g_{G, S}(n)=|{g ; l_S(g) £ n}| is called the growth function of the group G for the set of generators S. A non decreasing function t(n) has a subexponential growth if

lim n®¥  n Ö   t(n)   =1.

Theorem. Let

lim n®¥  n Ö   fG, S, k(n)   =1

for some natural k, then

lim n®¥  n Ö   gG, S(n)   =1.

This result allows us to prove that well-known p-groups of Gupta-Sidki and Gupta's group have intermediate growth.

Date received: October 1, 2005