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Injections of mapping class groups. (English) Zbl 1225.57001

The first main theorem of the paper shows that for all \(g \geq 2\) there exists \(h >g\) and an injective homomorphism \(\phi : \text{Mod}_g \rightarrow \text{Mod}_h\), where \(\text{Mod}_k\) denotes the mapping class group of a closed surface \(\Sigma_k\) of genus \(k\). These are examples of type-preserving homomorphisms, i.e. \(\phi\) maps pseudo-Anosov and reducible elements to pseudo-Anosov and reducible elements respectively. Further, for these examples, the authors obtain a \(\phi\)-equivariant isometric embedding of \(Teich(\Sigma_g)\) into \(Teich(\Sigma_h)\), where \(Teich\) denotes Teichmüller space. The techniques of proof are surprisingly elementary and clever.
For punctured surfaces, the authors construct even more interesting examples. They show that for all \(g \geq 2\) there exists \(h >g\) and an injective homomorphism \(\psi : \text{Mod}_{g,1} \rightarrow \text{Mod}_{h,1}\) such that \(\psi\) is not type-preserving. Here, \(\text{Mod}_{k,1}\) denotes the mapping class group of a surface \(\Sigma_{k,1}\) of genus \(k\) with one puncture. They further show that there is no \(\psi (\text{Mod}_{g,1})\)-invariant proper subset of \(\text{Teich}(\Sigma_{h,1})\) that is convex with respect to the Teichmüller metric. These examples therefore are very different in flavour from the first set of examples.
Reviewer: Mahan Mj (Howrah)

MSC:

57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
57M60 Group actions on manifolds and cell complexes in low dimensions
57R50 Differential topological aspects of diffeomorphisms
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