Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.

Putman 引入了分区曲面的概念，它是一个带有边界的曲面，其装饰限制了该曲面如何嵌入到更大的曲面中，并定义了分区曲面的 Torelli 群。在本文中，我们研究了划分曲面托雷利群的一些拓扑和动力学方面。更准确地说，首先我们获得划分曲面 Torelli 群的上同调维数的上界和下界，并表明当最多三个边界分量在边界划分中分组在一起时，这两个边界重合。其次，我们研究了相应曲线复合体上分分曲面 Torelli 群的渐近平移长度。我们证明最小渐近平移长度的渐近行为几乎类似于曲面的欧拉特征的倒数。这概括了第一作者和第二作者之前关于闭合曲面 Torelli 群的结果。