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 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 $$\ge 2$$ ≥ 2 ) with injective monodromy $$E\rightarrow B$$ E → 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.

中文翻译：

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映射类组、多个 Kodaira fibrations 和 CAT(0) 空间

我们研究了几个与小平纤维相关的几何和群理论问题，与更一般的黎曼曲面族以及逐面群相关。首先，我们提供了对以两种以上不同方式纤维化的小平纤维的限制，解决了 Catanese 和 Salter 关于它们存在的问题。然后我们证明，如果一个曲面上的曲面丛的基本群是 CAT(0) 群，则该丛必定具有单射性（除非单向性具有有限图像）。最后，给定一个在流形 B 上具有单射性 $$E\rightarrow B$$ E → B 的封闭黎曼曲面族（属 $$\ge 2$$ ≥ 2 ），我们解释了如何建立一个新的族黎曼曲面具有单射性，其基是总空间 E 的有限覆盖，其纤维具有更高的属。