arXiv subjects

Mehdi Yazdi

Publications and source records attributed to Mehdi Yazdi.

12 recordsLinked to original sources

arXiv2026

Thurston norm and the Euler class

In his influential work, Thurston introduced a norm on the second homology group of compact orientable 3-manifolds M, which by duality also determines a dual norm on the second cohomology group. A natural question, initiated by Thurston, is whether integral points on the boundary of the dual norm ball have a geometric interpretation. Thurston showed that the Euler class of the oriented tangent plane field to any taut foliation of M lies in the dual unit ball, and conjectured that, conversely, any integral point on the boundary of the dual unit ball is realised as the Euler class of a taut foliation. In this chapter, we discuss how several geometric, topological, and dynamical structures on a 3-manifold give rise to integral points in the dual unit ball of the Thurston norm, and what is known about Thurston's Euler class one conjecture in these contexts. These structures are taut foliations, tight contact structures, pseudo-Anosov flows, quasigeodesic flows, and circular orders on the fundamental group.

math.GT↗

arXiv2025

Bounds for the number of moves between pants decompositions, and between triangulations

Given two pants decompositions of a compact orientable surface $S$, we give an upper bound for their distance in the pants graph that depends logarithmically on their intersection number and polynomially on the Euler characteristic of $S$. As a consequence, we find an upper bound on the volume of the convex core of a maximal cusp (which is a hyperbolic structures on $S \times \mathbb{R}$ where given pants decompositions of the conformal boundary are pinched to annular cusps). As a further application, we give an upper bound for the Weil--Petersson distance between two points in the Teichmüller space of $S$ in terms of their corresponding short pants decompositions. Similarly, given two one-vertex triangulations of $S$, we give an upper bound for the number of flips and twist maps needed to convert one triangulation into the other. The proofs rely on using pre-triangulations, train tracks, and an algorithm of Agol, Hass, and Thurston.

math.GT↗

arXiv2025

Problem list on Foliations and Diffeomorphism groups

This document compiles problems proposed and discussed during the problem session at the conference Foliations and Diffeomorphism Groups (CIRM, 2024), organized by Hélène Eynard-Bontemps, Gaël Meigniez, Sam Nariman, and Mehdi Yazdi. The problems were contributed by participants and have been lightly edited by the organizers for clarity and coherence.

math.GT↗

arXiv2024

On $C^0$-stability of compact leaves with amenable fundamental group

In his work on the generalization of the Reeb stability theorem, Thurston conjectured that if the fundamental group of a compact leaf $L$ in a codimension-one transversely orientable foliation is amenable and if the first cohomology group $H^1(L;\mathbb{R})$ is trivial, then $L$ has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group $π_1(L)$ is amenable and $H^1(L;\mathbb{R})=0$, then for every transversely orientable codimension-one foliation $\mathcal{F}$ having $L$ as a leaf, there is a neighborhood of $\mathcal{F}$ in the space of $C^{1,0}$ foliations with Epstein $C^0$ topology consisting entirely of foliations that are locally a product $L \times \mathbb{R}$.

math.GT↗

arXiv2023

Thurston norm and Euler classes of tight contact structures

Bill Thurston proved that taut foliations of hyperbolic 3-manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realised as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this short note, we show that the previously constructed counterexamples for Euler classes of taut foliations in [Yaz20] are in fact realised as Euler classes of tight contact structures. This provides some evidence for the Euler class one conjecture for tight contact structures.

math.GT↗

arXiv2022

The computational complexity of knot genus in a fixed 3-manifold

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was known for rational homology 3-spheres, by the work of the first author.

math.GT↗

arXiv2021

Non-negative integral matrices with given spectral radius and controlled dimension

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an integral irreducible matrix with spectral radius $p$, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number $p$, there is an irreducible shift of finite type with entropy $\log(p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

math.DS↗

arXiv2020

On Thurston's Euler class one conjecture

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston's conjecture. Here counterexamples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

math.GT↗

arXiv2020

The fully marked surface theorem

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.

math.GT↗

arXiv2019

Lower bound for the Perron-Frobenius degrees of Perron numbers

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for biPerron numbers.

math.GT↗

arXiv2019

Pseudo-Anosov maps with small stretch factors on punctured surfaces

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. In particular it has been shown that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.

math.GT↗

arXiv2018

Lower Bound For Dilatations

We prove a new lower bound for the dilatation of an arbitrary pseudo-Anosov map on a surface of genus g with n punctures. Our bound improves the former super-exponential dependence on the genus by a polynomial dependence.

math.GT↗