[Paper Review] Geodesics in trees of hyperbolic and relatively hyperbolic groups

François Gautero|ArXiv.org|Oct 22, 2007

Geometric and Algebraic Topology17 references4 citations

TL;DR

This paper provides a detailed geometric analysis of geodesics in trees of hyperbolic and relatively hyperbolic groups using first-order geometric methods, avoiding reliance on isoperimetric inequalities. It establishes a new combination theorem for finite graphs of relatively hyperbolic groups under Farb’s and Gromov’s definitions, proving that semidirect products with free groups preserve relative hyperbolicity under suitable conditions.

ABSTRACT

We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb's and Gromov's definitions.

Motivation & Objective

• To provide a precise, first-order geometric description of geodesics in trees of hyperbolic and relatively hyperbolic groups.
• To establish a general combination theorem for finite graphs of relatively hyperbolic groups without relying on second-order isoperimetric methods.
• To unify the treatment of absolute and relative hyperbolicity by using quasi-geodesic approximation and the thin triangle property.
• To prove that semidirect products of relatively hyperbolic groups with free groups remain relatively hyperbolic under appropriate conditions.
• To offer a new, self-contained proof of the combination theorem in [3], treating both acylindrical and non-acylindrical cases.

Proposed method

• Use of telescopic quasi-geodesics with controlled vertical and horizontal segments to model geodesics in trees of spaces.
• Application of the thin triangle property and δ-hyperbolicity of strata to control horizontal deviations and ensure quasi-geodesic behavior.
• Modification of quasi-geodesics via sliding along vertical segments to achieve uniformly large vertical components, ensuring bounded neighborhood control.
• Use of quasi-isometric embeddings between strata to transfer geometric control across levels of the tree structure.
• Leveraging exponential separation of exceptional leaves in hyperbolic spaces to prevent return to central corridors.
• Adaptation of techniques from Bestvina-Feighn and Farb’s work to handle non-acylindrical actions and relative hyperbolicity.

Experimental results

Research questions

• RQ1 How can geodesics in trees of hyperbolic and relatively hyperbolic groups be precisely approximated using first-order geometric methods?
• RQ2 Under what conditions is the fundamental group of a finite graph of relatively hyperbolic groups itself relatively hyperbolic?
• RQ3 Can a combination theorem for relatively hyperbolic groups be established without relying on isoperimetric inequalities or 'black-box' results?
• RQ4 What conditions ensure that a semidirect product of a relatively hyperbolic group with a free group remains relatively hyperbolic?
• RQ5 How do exceptional leaves in hyperbolic spaces behave in terms of horizontal deviation and return to central corridors?

Key findings

• The paper provides a new, self-contained proof of the combination theorem for relatively hyperbolic groups, avoiding the use of isoperimetric inequalities.
• It proves that if $ G $ is weakly hyperbolic relative to $ ewmathcal{H} $, then $ G \rtimes \mathbb{F}_r $ is weakly hyperbolic relative to $ \newmathcal{H} $, and if $ G $ is strongly hyperbolic, then $ G \rtimes \mathbb{F}_r $ is strongly hyperbolic relative to a $ \mathbb{F}_r $-extension of $ \newmathcal{H} $.
• The proof relies on modifying quasi-geodesics by enlarging vertical segments to ensure bounded neighborhood control, using only the thin triangle property and quasi-isometric embeddings.
• The method successfully handles both acylindrical and non-acylindrical cases, including the case of mapping-tori of free group endomorphisms.
• For the case of surface groups with automorphisms induced by homeomorphisms, the resulting mapping-torus is shown to be weakly hyperbolic relative to boundary subgroups, maximal subsurface subgroups, and reduction curves.
• The exponential separation of exceptional leaves ensures that they do not return to the central corridor once deviating, which is crucial for maintaining quasi-convexity and hyperbolicity.