This is the second post intended to understand some of the ideas in Milnor’s “Note on curvature and the fundamental group.” This is the paper that introduces the idea of growth rates for groups and proves that the fundamental group in negative curvature has exponential growth (as well as a dual result on polynomial growth in nonnegative curvature). In the previous post, we described volume comparison results in negative curvature: we showed in particular that a curvature bounded above by $c < 0$ meant that the volumes of expanding balls grow exponentially in the radius. In this post, we’ll explain how this translates into a result about the fundamental group.

1. Growth rates of groups

Let ${G}$ be a finitely generated group, and let ${S}$ be a finite set of generators such that ${S^{-1} = S}$. We define the norm

$\displaystyle \left \lVert\cdot \right \rVert_S: G \rightarrow \mathbb{Z}_{\geq 0}$

such that ${\left \lVert g \right \rVert_S}$ is the length of the smallest word in ${S}$ that evaluates to ${g}$. We note that

$\displaystyle d(g, h) \stackrel{\mathrm{def}}{=} \left \lVert gh^{-1} \right \rVert_S$

defines a metric on ${G}$. The metric depends on the choice of ${S}$, but only up to scaling by a positive constant. That is, given ${S, S'}$, there exists a positive constant ${p}$ such that ${\left \lVert \cdot \right \rVert_S \leq p \left \lVert\cdot \right \rVert_{S'}}$. The metric space structure on ${G}$ is thus defined up to quasi-isometry. (more…)