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Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces


Llosa Isenrich,  Claudio
Max Planck Institute for Mathematics, Max Planck Society;

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Llosa Isenrich, C., & Py, P. (in press). Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces. Mathematische Annalen, Published Online - Print pending.

Cite as: http://hdl.handle.net/21.11116/0000-0007-E558-4
We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injective monodromy $E\to B$ over a manifold $B$, we explain how to build a new family of Riemann surfaces with injective monodromy whose base is a finite cover of the total space $E$ and whose fibers have higher genus. We apply our construction to prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of a closed surface of higher genus.