Embedding relatively hyperbolic groups into products of binary trees

TL;DR

This paper proves that relatively hyperbolic groups with virtually abelian peripherals can be quasiisometrically embedded into a product of binary trees, extending previous results for hyperbolic groups.

Contribution

It generalizes the embedding result from hyperbolic groups to a broader class of relatively hyperbolic groups with virtually abelian peripherals.

Findings

Relatively hyperbolic groups with virtually abelian peripherals embed into binary trees.

Develops a theory of diaries and linear statistics for embeddings.

Provides a framework to upgrade embeddings into infinite-valence trees to binary trees.

Abstract

We prove that if a group $G$ is relatively hyperbolic with respect to virtually abelian peripheral subgroups then $G$ quasiisometrically embeds into a product of binary trees. This extends the result of Buyalo, Dranishnikov and Schroeder in which they prove that a hyperbolic group quasiisometrically embeds into a product of binary trees. Inspired by Buyalo, Dranishnikov and Schroeder's Alice's Diary, we develop a general theory of diaries and linear statistics. These notions provide a framework by which one can take a quasiisometric embedding of a metric space into a product of infinite-valence trees and upgrade it to a quasiisometric embedding into a product of binary trees.