论文标题
涂料映射课程组的自动形态
Automorphisms of profinite mapping class groups
论文作者
论文摘要
对于$ s = s = s_ {g，n} $ a封闭式可区分的属$ g $的可区分表面，从中取出了$ n $点，因此$χ（s）= 2-2G-n <0 $，让$ \ m artrm {p Mathrm {p}γ（s）$ be $ s $ s $ s $ s $ s $ s＆s $ s $ s and $ \ mathrm {p} p} $ {p} p} p} $ \ mathrm {p} \checkγ（s）$分别是其涂鸦和一致性完成。后者可以用自然表示的图像$ \ mathrm {p} \wideHatγ（s）\ to \ operatorName {out}（{\widehatπ} _1（s））$，其中$ {\wideHatπ} _1（s）$是profinite copterals of profinite of pripinite of profinite of pripinite of saternite of sate $ sed $ sc $ s的$ s $。令$ \ operatorName {out}^{\ mathbb {i} _0}（\ mathrm {p} \wideHatγ（s））$和$ \ operatorName {out}^{out}^{\ mathbb {\ mathbb {i} _0} _0} _0}（\ mathrm）由非分离的dehn Twist生成的Procyclic子组的共轭类，让$ \ wideHat {\ permatatorName {gt}} $为涂鸦Grothendieck-TeichMüller组。然后，我们证明，对于$χ（s）<g-2 $，有一个自然的忠实表示：\ [\ [\ wideHat {\ propatatorName {gt}} \ hookrightArrow \ operatorArow \ operatorAtOrname {out}^{out}^{\ mathbb {\ mathbb {i} $σ_n$是$ s $的$ n $穿刺的对称组，一种自然的同构：\ [\ [\ perperatorname {out}^{\ mathbb {\ mathbb {i} _0}（\ mathrm {p}
For $S=S_{g,n}$ a closed orientable differentiable surface of genus $g$ from which $n$ points have been removed, such that $χ(S)=2-2g-n<0$, let $\mathrm{P}Γ(S)$ be the pure mapping class group of $S$ and $\mathrm{P}\widehatΓ(S)$ and $\mathrm{P}\checkΓ(S)$ be, respectively, its profinite and its congruence completions. The latter can be identified with the image of the natural representation $\mathrm{P}\widehatΓ(S)\to\operatorname{Out}({\widehatπ}_1(S))$, where ${\widehatπ}_1(S)$ is the profinite completion of the fundamental group of the surface $S$. Let $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))$ and $\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))$ be the groups of outer automorphisms which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist and let $\widehat{\operatorname{GT}}$ be the profinite Grothendieck-Teichmüller group. We then prove that, for $χ(S)<g-2$, there is a natural faithful representation: \[\widehat{\operatorname{GT}}\hookrightarrow\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\widehatΓ(S))\] and, letting $Σ_n$ be the symmetric group on the $n$ punctures of $S$, a natural isomorphism: \[\operatorname{Out}^{\mathbb{I}_0}(\mathrm{P}\checkΓ(S))\congΣ_n\times\widehat{\operatorname{GT}}.\]
