The barycenter method on singular spaces

Simplicial volume, Barycenter method, and Bounded cohomology

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many years. Finally we improve the degree theorem for $\mathbb Q$-rank $1$ locally symmetric spaces of Connell and Farb. We also address the issue of surjectivity of the comparison map in real rank $2$ case.

稳定立方剖分、双组合与重心

Stable cubulations, bicombings, and barycenters

本文证明映射类群与Teichmüller空间的层级凸包可被CAT(0)立方复形稳定逼近，进而推出二者分别具有半双曲性、粗等变双组合性及稳定粗重心，并推广至可着色层级双曲空间。