The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth

2007/07/05 by François Gautero, Gautero, Francois, Martin Lustig +1 · 2 citations
Mathematics · #20F65 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.0707.0822

openalex publication_date 2007/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the mapping torus group \FN \rtimes_α \Z of any automorphism α of a free group \FN of finite rank n ≥ 2 is weakly hyperbolic relative to the canonical (up to conjugation) family \mathcal H(α) of subgroups of \FN which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of α. Furthermore, we show that \FN \rtimes_α \Z is strongly hyperbolic relative to the mapping torus of the family \mathcal H(α). As an application, we use a result of Drutu-Sapir to deduce that \FN \rtimes_α \Z has Rapic Decay.

Citations

• Free-group automorphisms, train tracks and the beaded decomposition
• Geodesics in trees of hyperbolic and relatively hyperbolic groups

Cited by

• The geometry of subgroups of mapping tori of free groups
• Relatively irreducible free subroups in Out(\mathbbF)

Related

• Relative hyperbolicity of free-by-cyclic extensions
• PolExp growth for automorphisms of toral relatively hyperbolic groups
• Homology growth of polynomially growing mapping tori
• Hyperbolic automorphisms of free groups
• Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components