On the Growth of Groups and Automorphisms

We consider the growth functions βΓ(n) of amalgamated free products Γ = A ∗C B, where A ∼= B are finitely generated, C is free abelian and |A/C| = |A/B| = 2. For every d ∈ N there exist examples with βΓ(n) ' nβA(n). There also exist examples with βΓ(n) ' e. Similar behaviour is exhibited among Dehn functions. For Slava Grigorchuk, in friendship, with great respect. The first purpose of this note is to present explicit examples of groups which show that if one amalgamates two groups with polynomial growth of degree δ along isomorphic subgroups of index two, then the growth of the resulting group may be polynomial of degree δ + 1, polynomial of greater degree, or may be exponential. Similar jump behaviour is exhibited for Dehn functions of finite-index amalgams of both virtually abelian and virtually free groups. Let G be a group with finite generating set S. The growth function βG,S(n) counts the number of elements of G that can be expressed as a word of length at most n in the generators S and their inverses. If S ′ is another finite generating set for G then there exists a constant k > 0 such that βG,S(n) ≤ βG,S′(kn). We write f(n) g(n) when functions f, g : N → N are related by a constant in this way, and we write f(n) ' g(n) if, in addition, g(n) f(n). It is common to omit the subscript S from βG,S(n) and write βG(n) with the understanding that this function is only well-defined up to 'equivalence. Note that if H ⊂ G is a subgroup of finite index, then βH(n) ' βG(n). All finitely generated groups have growth e, because c ' e if c > 1, and if c > 2|S| then there are less than c reduced words of length n over the alphabet S±1. In his landmark paper [9], R. Grigorchuk proved that there exist finitely generated groups G of intermediate growth, i.e. e βG(n) e where 0 < δ < η < 1 (see also [10], [11]). On the other hand, the growth of every known finitely presented group is either ' e or else ' n for some d ∈ N. A celebrated theorem of M. Gromov [13] states that if βG(n) n, then G has a nilpotent subgroup G0 of finite index and βG(n) ' n where d ≤ p is an integer that can be calculated in terms of the ranks of the factors of lower central series of G0. See [12] for a wide-ranging survey of results concerning the growth of groups; also [14] Chap. VI. Let Γ = A ∗C B be an amalgamated free product of finitely generated groups. We assume that C has index at least 2 in both A and B. It follows immediately from the normal form theorem for amalgamated free products that Γ has exponential growth if |A/C| ≥ 3.