Lemma 7.4.19 (0PBA) · ScalingStacks

0PBA

Lemma 7.4.19. Given $\alpha,\beta\in R_{n}$ , we have $F_{L}(\langle\alpha,\beta\rangle)=\langle F_{R}(\alpha),F_{R}(\beta)\rangle$ and there is an injective morphism of groups $F_{\Gamma}:\Gamma_{n}\to\Gamma(S^{1}),\ (r,(l,\alpha))\mapsto(r,(F_{L}(l),F_{R}(\alpha)))$ .

Let $D$ be a subset of $\{1,\ldots,n\}\times\{\pm 1\}$ that embeds in its projection on $\{1,\ldots,n\}$ . Define $\partial:D\to T(S^{1})$ by $\partial((i,\nu_{i}))=c_{i}$ if $\nu_{i}=1$ and $\partial((i,\nu_{i}))=\iota(c_{i})$ otherwise. The morphism $F_{\Gamma}$ induces an isomorphism of groups $F_{D}:\Gamma_{D}\to\Gamma_{M}(S^{1},\partial(D))$ . We have $u<u^{\prime}$ if and only if $F_{D}(u)<F_{D}(u^{\prime})$ .

Let $\sigma$ be a map in ${\mathcal{S}}_{n}$ . We have $F_{R}(\llbracket\sigma\rrbracket)=\llbracket F(\sigma)\rrbracket$ , $m(F(\sigma))=F_{L}(m(\sigma))$ , $i(F(\sigma))=\ell(\sigma)$ and $\deg(F(\sigma))=F_{\Gamma}(\deg(\sigma))$ .

0PBB

Proof. Let $r,j\in\{1,\ldots,n\}$ and let $j^{\prime}\in{\mathbf{Z}}$ . We have

m_{c_{r}}(\llbracket F_{j}(j^{\prime}-j)\rrbracket)=|\{i\in r+n{\mathbf{Z}}\ |\ j\leq i<j^{\prime}\}|-|\{i\in r+n{\mathbf{Z}}\ |\ j>i\geq j^{\prime}\}|

and

m_{\iota(c_{r})}(\llbracket F_{j}(j^{\prime}-j)\rrbracket)=-|\{i\in r+n{\mathbf{Z}}\ |\ j<i\leq j^{\prime}\}|+|\{i\in r+n{\mathbf{Z}}\ |\ j\geq i>j^{\prime}\}|

In particular, $m_{c_{r}}(\llbracket F_{j}(1)\rrbracket)=\delta_{r,j}$ and $m_{\iota(c_{r})}(\llbracket F_{j}(1)\rrbracket)=-\delta_{r,j+1}$ . This shows that $F_{R}$ is injective. This shows also that given $i\in\{1,\ldots,n\}$ , we have

\langle\llbracket F_{i}(1)\rrbracket,\llbracket F_{j}(1)\rrbracket\rangle=(\delta_{i,j+1}+\delta_{i,j})F_{L}(\varepsilon_{j+1+n{\mathbf{Z}}})-(\delta_{i,j}+\delta_{i+1,j})F_{L}(\varepsilon_{j+n{\mathbf{Z}}})=F_{L}(\langle\alpha_{i+n{\mathbf{Z}}},\alpha_{j+n{\mathbf{Z}}}\rangle).

This shows the first equality and this shows that $F_{R}$ induces an injective morphism of groups $F_{\Gamma}$ .

Taking quotients, we obtain an injective morphism of groups $F_{D}:\Gamma_{D}\to\Gamma(S^{1},\partial(D))$ compatible with the order and with image $\Gamma_{M}(S^{1},\partial(D))$ .

Consider $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J)$ . Given $d\in{\mathbf{Z}}$ , we have $\llbracket F_{r}(d)\rrbracket=F_{R}(\alpha_{r,r+d})$ , hence $F_{R}(\llbracket\sigma\rrbracket)=\llbracket F(\sigma)\rrbracket$ .

We have

m(F(\sigma))=\sum_{r,j\in\tilde{I}\cap[1.n]}(m_{c_{r}}-m_{\iota(c_{r})})(\llbracket F_{j}(\sigma(j)-j)\rrbracket)e_{c_{r}}

and

m(\sigma)=\sum_{r,j\in\tilde{I}\cap[1,n]}\alpha_{j,\sigma(j)}\cdot\varepsilon_{r+n{\mathbf{Z}}}.

Since

\alpha_{j,\sigma(j)}\cdot\varepsilon_{r+nZ}=(m_{c_{r}}-m_{\iota(c_{r})})(F(\sigma))\varepsilon_{r+nZ},

it follows that $m(F(\sigma))=F_{L}(m(\sigma))$ .

Consider $i_{1},i_{2}\in\tilde{I}$ with $0\leq i_{1}<i_{2}<n$ . We have $i(F(\sigma)_{a_{i_{1}}},F(\sigma)_{a_{i_{2}}})=i(\gamma_{1},\gamma_{2})$ for some minimal paths $\gamma_{l}$ in $F(\sigma)_{a_{i_{l}}}$ by Lemma 7.3.22. Lemma 6.2.3 shows that $i(F(\sigma)_{a_{i_{1}}},F(\sigma)_{a_{i_{2}}})=\bigl|{\lfloor\frac{\sigma(i_{2})-\sigma(i_{1})}{n}\rfloor}\bigr|$ . Lemma 6.2.2 shows now that $i(F(\sigma))=\ell(\sigma)$ . ∎