Two new constructions in the theory of projection complexes
Abstract
In this paper we provide two new constructions that are useful for the theory of projection complexes developed by Bestvina, Bromberg, Fujiwara and Sisto. We prove that there exists a subtree of the projection complex which is quasiisometric to the projection complex. We use this subtree to form a tree of metric spaces, which is a subgraph of the quasi-tree of metrics spaces and quasiisometric to it. These constructions simplify the metric structure (up to quasiisometry) of the projection complex and the quasi-tree of metric spaces. As an application, we use these constructions to provide a shorter proof of Hume's theorem that the mapping class group admits a quasiisometric embedding into a finite product of trees.
Keywords
Cite
@article{arxiv.2404.02730,
title = {Two new constructions in the theory of projection complexes},
author = {Patrick S. Nairne},
journal= {arXiv preprint arXiv:2404.02730},
year = {2024}
}
Comments
The previous version of this preprint has been split into two parts. This version forms the first part. The second part, which concerns relatively hyperbolic groups, will be uploaded to arXiv soon