arXiv subjects

Ahmad Rafiqi

Publications and source records attributed to Ahmad Rafiqi.

4 recordsLinked to original sources

arXiv2025

A Simple Computation of Teichmüller Polynomials from Integer Permutations

We present a simple method to compute the Teichmüller polynomial of the fibered face of a hyperbolic $3$-manifold $M_ϕ$ obtained as the mapping torus of a pseudo-Anosov homeomorphism $ϕ$ of a closed surface. We assume $ϕ$ has orientable invariant foliations and fixes each singular trajectory. We use a characterisation of such homeomorphisms in terms of a permutation of a finite set of integers to give a direct implementation of McMullens algorithm using train tracks. Train tracks with a single vertex suffice in this case. As an application, for each $p\in\mathbb{Z}_{\geq0}$, we find an infinite sequence of Teichmüller polynomials $Θ_{g,p}$ associated to pseudo-Anosov maps on surfaces of genus $g\geq2$, such that the hyperbolic 3-manifold obtained as the mapping torus has first Betti number $g$. These polynomials realize a positive proportion of bi-Perron units of each degree as pseudo-Anosov stretch-factors.

math.GT↗

arXiv2024

Creating pseudo-Anosov Maps from Permutations and Matrices

We provide an integral combinatorial characterization of pseudo-Anosov maps on closed oriented surfaces of genus g > 1. We show that an orientation-preserving pseudo-Anosov homeomorphism with orientable foliations and fixing all critical trajectories can be encoded as a permutation of 2g+v-1 positive integers, where v is the number of singular points of the foliations (disregarding multiplicity). We call such a permutations an ordered block permutation (OBP), and it satisfies an admissiblity condition. Conversely, we show that a surface along with measured foliations (up to scaling) and the pseudo-Anosov map can be uniquely constructed out of the data of an admissible permutation of 2g+v-1 positive integers. In particular, for closed surfaces, we construct every orientable foliation invariant under a pseudo-Anosov homeomorphism.

math.DS↗

arXiv2016

Is a typical bi-Perron number a pseudo-Anosov dilatation?

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area one abelian differentials for low genus cases.

math.GT↗

arXiv2015

Constructing pseudo-Anosov maps with given dilatations

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

math.GT↗