arXiv research

New stretch maps minimize distortion in geometric group theory.

Study stretch laminations in hyperbolic 3-manifolds via circle-valued maps.

Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.

Minimal stretch factors for certain pseudo-Anosov maps are bounded.

We explicitly construct pseudo-Anosov maps on the closed surface of genus $g$ with orientable foliations whose stretch factor $λ$ is a Salem number with algebraic degree $2g$. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree $d$, for each positive even integer $d$ s…

Lower bound on stretch factor for periodic maps.

In 1974, Thurston proved that, up to isotopy, every automorphism of closed orientable surface is either periodic, reducible, or pseudo-Anosov. The latter case has lead to a rich theory with applications ranging from dynamical systems to low dimensional topology. Associated with every pseudo-Anosov map is a real number …

The paper studies maximal stretch and Lipschitz maps on negatively curved manifolds.

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…

The period map for 4-manifolds is dense and surjective under certain conditions.

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, …

Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface $S$ can be as high as the dimension of the Teichmüller space of $S$. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…

Study connects group invariants through outer automorphisms and polynomial relations.

We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in…

The study finds all trace field degrees for Torelli group mappings.

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured la…

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the …

The paper studies pseudo-Anosov maps from typical Thurston constructions.

We give a new description of the Arnoux-Yoccoz mapping classes as a product of two Dehn twists and a finite order element. The construction is analogous to Penner's construction of mapping classes with small stretch factors.

We introduce a construction of pseudo-Anosov homeomorphisms on n-times punctured spheres and surfaces with higher genus using only sufficiently many positive half-twists. These constructions can produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors, i…

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

In the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense o…

For every irreducible automorphism $φ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $φ$, whose stretch factor is the product of the expanding eigenva…

Maps between acute triangles with minimal stretch found and studied.

We introduce a property of mutation loops, called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropical $\mathcal{X}$-transformation. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with that of a pseudo-Anosov mapping clas…

We study Thurston's Lipschitz and curve metrics, as well as the arc metric on the Teichmueller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston'…

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$-trees, using stretching factors of $G$-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…

We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construc…

Unique geodesics selected by energy minimization in Teichmüller space.

For a compact manifold with boundary $X$ we introduce the $n$-fold scattering stretched product $X^n_{\text{sc}}$ which is a compact manifold with corners for each $n,$ coinciding with the previously known cases for $n=2,3.$ It is constructed by iterated blow up of boundary faces and boundary faces of multi-diagonals i…

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

Finsler metrics with relatively non-negative (non-positive, respectively), constant and isotropic stretch curvatures are investigated in this paper. In particular, it is proved that every non-Riemannian $(α, β)$-metric with a nonzero constant flag curvature and a non-zero relatively isotropic stretch curvature over a m…

The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us…

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

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 …

Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.

Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic l…

The Teichmüller space $\mathcal{T}(Σ)$ of a surface $Σ$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(Σ)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.

Study finds how periodic surfaces can bend without stretching.

Minimal stretch factor for non-orientable surfaces is small.

We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer $n \geq 2$. In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT represen…

Study how large-scale flows align small-scale vortices in 3D Euler equations.

Study on solvable Lie groups with specific Weyl connections.

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who…