ScalingStacks

Lemma 6.2.3. Let $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ where $I=\{i_{1}+n{\mathbf{Z}},i_{2}+n{\mathbf{Z}}\}$ and $J=\{j_{1}+n{\mathbf{Z}},j_{2}+n{\mathbf{Z}}\}$ with $1\leq i_{1}\neq i_{2}\leq n$, $1\leq j_{1}\neq j_{2}\leq n$ and $\sigma(i_{r})=j_{r}\pmod{n}$ for $r\in\{1,2\}$. Fix $\beta:\{i_{1},i_{2},j_{1},j_{2}\}\to{\mathbf{R}}$ increasing with $|\beta(u)-\beta(v)|<1$ for all $u,v$.

Consider $\gamma_{r}:[0,1]\to{\mathbf{R}}$ continuous with $\gamma_{r}(0)=\beta(i_{r})$ and $\gamma_{r}(1)=\beta(j_{r})+\frac{\sigma(i_{r})-j_{r}}{n}$ for $r\in\{1,2\}$. We have

$$\ell(\sigma)\leq|\{t\in[0,1]\ |\ e^{2i\pi\gamma_{1}(t)}=e^{2i\pi\gamma_{2}(t)}\}|$$

with equality if, for all $r\in\{1,2\}$, the map $\gamma_{r}$ is affine.