arXiv research

Every infinitely edge-connected graph has a minor of Farey graph or $T_{\aleph_0}\ast t$.

Paper defines Farey Recursive Functions and explores their properties.

We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…

Connected graph for twice-punctured torus curves.

Study geodesics in 3-torus, determining complements' topology.

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the $k$-Farey graphs $\mathcal{F}_k$ and $\mathcal{F}_{\leqslant k}$, two natural variants of the Farey graph in which we relax the edge condition to indicate intersection number $=k$ or $\le k$, respectively. The former, $\…

Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.

Paper develops geometry for Kleinian groups using Farey polynomials.

Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.

Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the 1-skeleton of the pants complex.

Study of SU(2,1) character varieties on one-holed torus.

The paper studies algebraic integer relations and sequences converging to 4.

Study of $q$-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.

The paper generalizes Farey tessellation to 3D hyperbolic space.

Study calculates stable norm of slit tori using Farey sequence.

The Farey tree helps embed rational balls and lens spaces into complex projective space.

Spheres in curve graphs are connected, proving Gromov boundary linearity.

New patterns deform Farey triangulation in symmetric space.

Optimal Farey sequence for $Γ_0(2^n)$ with upper bound $2^{n-1}$.

We describe the (P)SL(2,C) character varieties of all 2-bridge knots and the diagonal character varieties for all 2-bridge links in terms of a set of polynomials defined using Farey recursion.

Study quasisymmetric maps on hyperbolic plane boundaries.

Octagon map accelerates diagonal changes algorithm.

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

We use the geometry of the Farey graph to give an alternative proof of the fact that if $A \in GL_2\mathbb Z$ and $G_A=\mathbb Z^2 \rtimes_A \mathbb Z$ is generated by two elements, there is a single Nielsen equivalence class of $2$-element generating sets for $G_A$ unless $A$ is conjugate to $\pm \left(\begin {smallma…

A new proof shows how to characterize maps using simple geometry.

The study of Farey polynomials connects geometry, topology, and combinatorics.

Study Agol cycles in pseudo-Anosov 3-braids.

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…

Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.

Study of circle homeomorphisms with square summable diamond shears.

Study jigsaw constructions of hyperbolic lattices and answer questions on arithmeticity and pseudomodularity.

Geodesic patterns, shears, and Anosov representations of the modular group.

For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant w…

Let $K$ be a nontrivial knot in $S^{3}$ and $t(K)$ its tunnel number. For any $(p\geq 2,q)$-slope in the torus boundary of a closed regular neighborhood of $K$ in $S^{3}$, denoted by $K^{\star}$, it is a nontrivial cable knot in $S^{3}$. Though $t(K^{\star})\leq t(K)+1$, Example 1.1 in Section 1 shows that in some cas…

We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. F…

We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of pe…

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property th…

A musical instrument based on moduli spaces lets users hear geometric concepts.

Proves effective slope gaps for lattice surfaces.

The paper proves a free product decomposition for congruence subgroups with constraints.

New tree structure for pseudo-Anosovs from interval maps.

Classifies positive integral friezes on surfaces.

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…

Identifies half-space neighborhoods of pleating rays in the Riley slice of Schottky groups.

This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…

Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.