Asymmetry of outer space - injectivity radius

Question

I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina.

In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact that the injectivity radius $\text{injrad}(x_k)$ of $x_k$ goes to $0$, and in fact $d(x_k,x_2)\sim -\log{\text{injrad}(x_k)}$".

My problem is that I don't know what injectivity radius is. Neither in the specific case of $CV_n$, nor generally.

Could someone explain or provide some relevant material to study?

Thanks in advance!

Answer 1

Given a geodesic metric space $X$ and a point $p\in X$, the injectivity radius $\mathrm{injrad}(p)$ is the maximum value of $r$ such that every point in the open ball $B(p,r)$ is connected to $p$ by a unique geodesic. Injectivity radius is important in the study of Riemannian manifolds (where it is often defined in terms of the exponential map) but the definition makes sense for arbitrary geodesic metric spaces.

Here are some examples:

1. If $X$ is a unit circle, then $\mathrm{injrad}(p)=\pi$ for all $p\in X$. The same holds true for the unit sphere of any dimension.

2. If $X=\mathbb{R}^n$ under the Euclidean metric, then $\mathrm{injrad}(p)=\infty$ for every point $p\in\mathbb{R}^n$, since every pair of points in $\mathbb{R}^n$ is connected by a unique geodesic.

3. Let $X\subseteq \mathbb{R}^2$ be the union of the unit circle and the line $y=1$. Then $\mathrm{injrad}(p)=\pi$ for every point $p$ on the unit circle and $\mathrm{injrad}((x,1))=\pi + |x|$ for every point $(x,1)$ on the line $y=1$.

4. Let $X\subseteq\mathbb{R}^2$ be the Hawaiian earring, i.e. the union of circles of radius $1/n$ centered at $(1/n,0)$ for $n\in\mathbb{N}$. Then $\mathrm{injrad}((0,0))=0$ and more generally $\mathrm{injrad}(p)=d(p,(0,0))$ for all $p\in X$.