Stable cylinders and fine structures for hyperbolic groups and curve graphs

TL;DR

This paper proves that certain hyperbolic groups and curve graphs possess globally stable cylinders and fine structures, using advanced geometric constructions, and demonstrates their embeddings into finite products of quasitrees.

Contribution

It establishes the existence of stable cylinders for all residually finite hyperbolic groups and curve graphs, and introduces a method to improve hyperbolic space properties via a generalized Sageev construction.

Findings

Hyperbolic groups admit globally stable cylinders.

Curve graphs can be embedded in finite products of quasitrees.

The approach enhances hyperbolic space properties with fine structures.

Abstract

In 1995, Rips and Sela asked if torsionfree hyperbolic groups admit globally stable cylinders. We establish this property for all residually finite hyperbolic groups and curve graphs of finite-type surfaces. These cylinders are fine objects, and the core of our approach is to upgrade the hyperbolic space to one with improved fine properties via a generalisation of Sageev's construction. The methods also let us prove that curve graphs of surfaces admit equivariant quasiisometric embeddings in finite products of quasitrees.