Growth of finitely generated solvable groups and curvature of Riemannian manifolds

TLDR

For a finitely generated group Γ, the growth function g_s(m) counts the number of distinct elements representable by words of length less than m over a finite generating set. Milnor proved that the asymptotic growth of g_s is independent of the generating set and that curvature bounds on a Riemannian manifold M translate into corresponding bounds on the growth of its fundamental group π^M, with the paper detailing the specific types of such bounds.

Abstract

If a group Γ is generated by a finite subset 5, then one has the gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J. Milnor [9] shows that the asymptotic behaviour of gs does not depend on choice of finite generating set S c Γ, and that lower (resp. upper) bounds on the curvature of a riemannian manifold M result in upper (resp. lower) bounds on the growth function of π^M). The types of bounds on the growth function are