Stable cylinders and fine structures for hyperbolic groups and curve graphs

TL;DR

The paper addresses the existence of globally stable cylinders in torsion-free hyperbolic groups as well as in curve graphs, extending the framework to all residually finite hyperbolic groups and finite-type surface curve graphs. It introduces a generalized Sageev duality to produce thickened hyperbolic spaces with fine median-like properties and develops a strong (QT) framework: hyperbolic spaces embed equivariantly and quasi-isometrically into finite products of quasitrees. This enables the demonstration that residually finite hyperbolic groups admit globally stable cylinders and that curve graphs admit equivariant quasiisometric embeddings into finite products of quasitrees, leading to strong (QT) for the pair $(\mathcal{C}\Sigma, \mathrm{MCG}(\Sigma))$ and globally stable cylinders for curve graphs. The results unify techniques across hyperbolic groups, curve graphs, and mapping class groups, with applications to Artin groups of large hyperbolic type via coned-off Deligne complexes, offering new tools for solving equations in hyperbolic groups and for understanding large-scale geometric structure in these settings.

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.

Figures (3)

• Figure 1: The stability condition for cylinders.
• Figure 2: The case in the proof of Proposition \ref{['prop:0-sep_quasitree']} where many elements of $c$ separate $w$ from $h_1$ and $h_2$.
• Figure 3: Proof of Proposition \ref{['prop:few_projections']}. The yellow walls $k_1$ and $k_2$ separate $h_2$ from both $y$ and $z$. Hence no other yellow wall can both cross $h_2$ and separate $y$ from $z$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: petytzalloum:constructing
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 2.5