Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length

2015/10/04 by Tarik Aougab, Aougab, Tarik, Samuel J. Taylor +1
Computer Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.GT

paper · pdf · doi:10.48550/arxiv.1510.00995

12 page, 1 figure

arxiv created 2015/10/04 · openalex publication_date 2015/10/04 · arxiv updated 2015/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given ϕ a pseudo-Anosov map, let ℓ_T(ϕ) denote the translation length of ϕ in the Teichmüller space, and let ℓ_C(ϕ) denote the stable translation length of ϕ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic χ(S), the minimal possible ratio τ(ϕ) = (ℓ_T(ϕ))/(ℓ_C(ϕ)) is log(|χ(S)|), up to uniform additive and multiplicative constants. In this short note, we introduce a new construction of such ratio optimizers and demonstrate their abundance in the mapping class group. Further, we show that ratio optimizers can be found arbitrarily deep into the Johnson filtration as well as in the point pushing subgroup.

Related

• On Translation Lengths of Pseudo-Anosov Maps on the Curve Graph
• Comparing Teichmüller and curve graph translation lengths
• Sequences of pseudo-Anosov mapping classes and their asymptotic behavior
• Asymptotics for pseudo-Anosov elements in Teichmuller lattices
• On the minimum of asymptotic translation lengths of point-pushing pseudo-Anosov maps on one punctured Riemann surfaces