Topological and dynamical properties of Torelli groups of partitioned surfaces

TL;DR

This paper investigates the topological and dynamical properties of Torelli groups associated with partitioned surfaces, providing bounds on their cohomological dimension and analyzing their action on curve complexes, extending known results to surfaces with boundary.

Contribution

It introduces bounds on the cohomological dimension of Torelli groups for partitioned surfaces and studies their asymptotic translation lengths, generalizing previous results to surfaces with boundary.

Findings

Bounds on cohomological dimension coincide for up to three boundary components.

Asymptotic translation length behaves like the reciprocal of the Euler characteristic.

Results extend Torelli group properties from closed to partitioned surfaces with boundary.

Abstract

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.