Let $S_{g,k}$ be a genus $g$ surface with $k$ punctures. Let $\mathrm{Mod}(S)$ be the extended mapping class group of a surface, defined as the isotopy classes of the self-homeomorphisms of surface $S$. Let $\mathcal{T}(S_{g,k})$ be the Teichmüller space of $S$. Suppose there is an injective map $\phi : \mathrm{Mod}(S_{g,k}) \to \mathrm{Mod}(S_{g',k'})$, then how do I show that there is a continuous $\phi$-equivariant map $\Phi : \mathcal{T}(S_{g,k}) \to \mathcal{T}(S_{g',k'})$? Furthermore, if there is a surjective homomorphism from $\mathrm{Mod}(S_{g',k'})$ to $\mathrm{Mod}(S_{g,k})$, then how does one show that $\Phi$ is an isometric embedding with respect to Teichmüller metric? I have studied mapping class groups. I also know some important results about Teichmüller space, but I don't have a thorough understanding of Teichmüller space. I am currently studying the 2018 paper "Injections of mapping class groups" by Aramayona, Leininger and Souto (arXiv: https://arxiv.org/abs/0811.0841).
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$\begingroup$ Why do you think these results are true? How much of Teichmuller theory do you know? $\endgroup$– Moishe KohanCommented Apr 8, 2023 at 14:32
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$\begingroup$ @MoisheKohan The first claim, I think, follows from the fact that the mapping class group is the fundamental group of the moduli space and the fact that the mapping class group is generated by finite-order elements. I have no idea how to prove the second claim. I am using "A Primer on mapping class groups" for reference. Up until now, I have only read the statements of the main theorems and propositions from the book. $\endgroup$– ASHCommented Apr 8, 2023 at 17:49
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$\begingroup$ @MoisheKohan Can you suggest references in the right direction? $\endgroup$– ASHCommented Apr 8, 2023 at 17:59
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