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Jonah Gaster

Publications and source records attributed to Jonah Gaster.

At least 19 recordsLinked to original sources
arXiv2025

From arcs to curves: quadratic growth of 1-systems

We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ grows quadratically in $|\chi|$. This resolves a longstanding question of Farb-Leininger, up to multiplicative constants. Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of \textit{almost nibs}, \textit{flowers}, and \textit{stem systems} in order to account for how certain polygons built from pairs of curves in the collection distribute their area over the surface.

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arXiv2023

Twist numbers on hyperbolic once-punctured tori

On a hyperbolic surface homeomorphic to a torus with a puncture, each oriented simple geodesic inherits a well-defined relative twist number in $[0,1]$, given by the ratio to its hyperbolic length of the hyperbolic distance between the orthogonal projections of the cusp (or boundary) on its left and right, respectively. With the Markov Uniqueness Conjecture in mind, the twist numbers for simple geodesics on the modular torus $\mathcal{X}$ are of particular interest. Up to isometries of $\mathcal{X}$, simple geodesics are parameterized naturally by $\mathbb{Q}\cap[0,1]$, and the relative twist number yields a map $\tau_{\mathcal{X}}:\mathbb{Q}\cap [0,1] \to [0,1]$. We use hyperbolic geometry and the Farey graph to show that the graph of $\tau_{\mathcal{X}}$ is dense in $[0,1]\times [0,1]$, and the same conclusion holds for any complete hyperbolic structure $\mathcal{Y}$ on the punctured torus. It follows that the twist number of a simple closed curve on the punctured torus does not extend continuously to the space of measured laminations. We also include some explicit calculations of geometric quantities associated to Markov triples, and the curious fact that $\tau_{\mathcal{X}}$ is never equal to zero.

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arXiv2022

Combinatorially random curves on surfaces

We study topological properties of random closed curves on an orientable surface $S$ of negative Euler characteristic. Letting $\gamma_{n}$ denote the conjugacy class of the $n^{th}$ step of a simple random walk on the Cayley graph driven by a measure whose support is on a finite generating set, then with probability converging to $1$ as $n$ goes to infinity, (1) the point in Teichm\"uller space at which $\gamma_{n}$ is length-minimized stays in some compact set; (2) the self-intersection number of $\gamma_{n}$ is on the order of $n^{2}$, the minimum length of $\gamma_{n}$ taken over all hyperbolic metrics is on the order of $n$, and the metric minimizing the length of $\gamma_{n}$ is uniformly thick; and (3) when $S$ is punctured and the distribution is uniform and supported on a generating set of minimum size, the minimum degree of a cover to which $\gamma_{n}$ admits a simple elevation (which we call the $\textit{simple lifting degree}$ of $\gamma_{n}$) grows at least like $n/\log(n)$ and at most on the order of $n$. We also show that these properties are $\textit{generic}$, in the sense that the proportion of elements in the ball of radius $n$ in the Cayley graph for which they hold, converges to $1$ as $n$ goes to infinity. The lower bounds on simple lifting degree for randomly chosen curves we obtain significantly improve the previously best known bounds which were on the order of $\log^{(1/3)}n$. As applications, we give relatively sharp upper and lower bounds on the dilatation of a generic point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of its defining curve, as well as upper bounds on the simple lifting degree of a random curve in terms of its intersection number which outperform bounds for general curves.

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arXiv2021

Boundary slopes for the Markov ordering on relatively prime pairs

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to $q$, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus.

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arXiv2020

Curves on the torus intersecting at most k times

We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$, and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt{k} \log k)$. By contrast, our methods are elementary, combinatorial, and geometric.

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arXiv2020

The sum of Lagrange numbers

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio.

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arXiv2020

Good stackings, bislim structures, and invariant staggerings

Two seemingly different properties of 2-complexes were developed concurrently as criteria for the nonpositive immersion property: 'good stackings' and 'bislim structures'. We establish an equivalence between these properties by introducing a third property 'invariant staggerings' mediating between them.

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arXiv2020

A short proof of a conjecture of Aougab-Huang

In response to Sanki-Vadnere, we present a short proof of the following theorem: a pair of simple curves on a hyperbolic surface whose complementary regions are disks has length at least half the perimeter of the regular right-angled $(8g-4)$-gon.

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arXiv2019

Algebraic k-systems of curves

A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in \Delta $ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of curves on a surface of genus $g$ is $2g+1$ when $g\ge 3$ or $k$ is odd, and $2g$ otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as $g^2$.

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arXiv2019

Computing harmonic maps between Riemannian manifolds

In the previous paper [GLM2018], we showed that the theory of harmonic maps between Riemannian manifolds may be discretized by introducing triangulations with vertex and edge weights on the domain manifold. In the present paper, we study convergence of the discrete theory to the smooth theory when taking finer and finer triangulations. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps. Our computer software Harmony implements these methods to computes equivariant harmonic maps in the hyperbolic plane.

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arXiv2018

Computing discrete equivariant harmonic maps

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

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arXiv2018

Bicollapsibility and groups with torsion

We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that many such groups act properly and cocompactly on a CAT(0) cube complex.

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arXiv2018

Combinatorics of $k$-Farey graphs

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, $\mathcal{F}_k$, is disconnected when $k>1$. In fact, we find that the number of connected components is infinite if and only if $k$ is not a prime power. Moreover, we find that each component of $\mathcal{F}_k$ is an infinite-valence tree whenever $k$ is even, and $\mathrm{Aut}(\mathcal{F}_k)$ is uncountable for $k>1$. As for $\mathcal{F}_{\leqslant k}$, Agol obtained an upper bound of $1+\min\{p:p\text{ is a prime}>k\}$ for both chromatic and clique numbers, and observed that this is an equality when $k$ is either one or two less than a prime. We add to this list the values of $k$ that are three less than a prime equivalent to $11\ (\mathrm{mod}\ 12)$, and we show computer-assisted computations of many values of $k$ for which equality fails.

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arXiv2016

The random graph embeds in the curve graph of any infinite genus surface

The random graph is an infinite graph with the universal property that any embedding of $G-v$ extends to an embedding of $G$, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface $Σ$ if and only if $Σ$ has infinite genus, showing that the curve system on an infinite genus surface is "as complicated as possible".

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arXiv2016

Packing curves on surfaces with few intersections

Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|χ(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \mathcal{N}_{k}(S) =O \left( \frac{ |χ|^{3k}}{ ( \log |χ| )^2 } \right) , $$ In particular, the size of a maximal 1-system grows sub-cubically in $|χ(S)|$. The proof uses a circle packing argument of Aougab-Souto and a bound for the number of curves of length at most $L$ on a hyperbolic surface. When the genus $g$ is fixed and the number of punctures $n$ grows, we can improve our estimates using a different argument to give $$ \mathcal{N}_{k}(S) \leq O(n^{2k+2}) . $$ Using similar techniques, we also obtain the sharp estimate $\mathcal{N}_{2}(S)=Θ(n^3)$ when $k=2$ and $g$ is fixed.

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arXiv2016

On the complexity of finite subgraphs of the curve graph

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the 'nested complexity length', which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large half-graphs do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large multipartite subgraphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

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arXiv2016

Coloring curves on surfaces

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.

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