Other Growth Conditions
Alexander Lubotzky, Dan Segal · Birkhäuser Basel eBooks · 2003
Subgroup growth is one way to measure the growth of finite images of a group. Of course there are other, equally natural, ways to do this. One that has received a certain amount of attention is known (not quite accurately) as ‘index growth’: that is, the growth of |G : G n | as a function of n. Novikov and Adian showed that this index is in general infinite, even for finitely generated groups G (the negative solution of the original Burnside problem); as we are concerned here primarily with finite quotients, we concentrate rather on $$|\hat{G}:{{\hat{G}}^{n}}| = \sup \left\{ {|\tilde{G}:{{{\tilde{G}}}^{n}}| :\tilde{G} {\text{a}} {\text{finite}} {\text{quotient}} {\text{of}} G} \right\},$$ (here $${{\hat{G}}^{n}}$$ denotes the closed subgroup generated by all nth powers in the profinite completion of G). The positive solution of the restricted Burnside problem by Zelmanov shows that if G is finitely generated then this number is indeed finite, for every n. However, it can grow exceedingly fast a multiply-iterated exponential function of n (see [Vaughan-Lee & Zelmanov 1999], §2). Thus the following should be a strong restriction: a group G has polynomial index growth, or PIG, if there exists γ>0 such that |Ĝ : Ĝ n | ≤ nγ for all n.