arXiv research

Authors prove quantum invariant conjecture for figure-eight knot complement.

We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ 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 …

Study on hyperbolic triangles and once-punctured torus groups, focusing on group relations and deformations.

Study of harmonic maps to the circle with applications to hyperbolic 3-manifolds.

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures: A reducible 3-manifold admits an Anosov torus if an…

The volume conjecture is extended for surface diffeomorphisms with quantum invariants.

Upper bound on 3-manifold volumes from surface homeomorphisms.

Geometrically, twist numbers on punctured tori are dense and non-continuous.

The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.

Study character varieties of hyperbolic 3-manifolds using bundle methods.

We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüll…

Study of transitivity in partially hyperbolic maps with expanding linear part.

We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …

The study finds hyperbolic twisted torus links for certain twists.

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations a…

We extend a theorem of Masur and Wolf which says that given a hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four-holed sphere, the one-holed torus, a…

Geodesic flow on orbifolds has a genus 1 Birkhoff section.

A new method computes Teichmüller polynomials from integer permutations.

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…

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a large…

In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps f…

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use t…

Non-coherence proven for certain groups with specific mapping tori.

Metric WPD for pseudo-Anosov maps shows many unbounded quasi-morphisms.

Study of flows on 7D manifolds with holomorphic properties.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

Hyperbolic links in thickened torus decompose into angled tetrahedra.

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'…

The study of pseudo-Anosovs through mapping torus geometry.

Paper shows minimum 10 vertices for hyperbolic origami 2-torus.

Let $Σ$ be a hyperbolic surface. We study the set of curves on $Σ$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $γ_0$. For example, in the particular case that $Σ$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $γ_0$ and of at most l…

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

New quasimorphisms show stable commutator lengths are not equivalent.

We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have $C^1$ robustly transitive endomorphisms with critical points on surfaces, and the only surfaces …

We show that if a hyperbolic 3-manifold $M$ with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then $M$ is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifold contains an essential torus which meets t…

We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism…

In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as …

Formula estimates pseudo-Anosov maps' fixed points, linking to surface properties.

Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.

The paper calculates intertwiners for a torus and proves a conjecture about their limits.

Upper and lower bounds for hyperbolic rod complements in 3-torus volumes.

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded …

This research studies end-periodic mapping tori and their hyperbolic structures.

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…

In this note we announce several results concerning the SL(2,C) character variety ${\mathcal X}$ of the one-holed torus. We give a description of the largest open subset ${\mathcal X}_{BQ}$ of ${\mathcal X}$ on which the mapping class group $Γ$ acts properly discontinuously, in terms of two very simple conditions, and …