arXiv subjects

Tarik Aougab

Publications and source records attributed to Tarik Aougab.

At least 19 recordsLinked to original sources

arXiv2026

From arcs to curves: quadratic growth of 1-systems

We show that a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ has at most $2016|χ|^2+338|χ|$ curves. Up to multiplicative constants, this resolves a thirty-year old problem (see Problem 2.12(b) from the K3 Problem List). Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of tulips, flowers, and stem systems in order to account for how certain polygons built from pairs of curves in the collection distribute area over the surface.

math.GT↗

arXiv2026

Navigating the AI crisis: a humble guide for students

We present some basic recommendations for (mostly graduate) students when it comes to choosing to use, or avoid, AI in their mathematical research and education. These include: maintaining a connection to the tried and true ways for learning mathematics; focusing on communication and the sharing of ideas with other people; thinking critically about the politics of labor and organizing with your cohort and others at your home institution; and being in touch with the realities of the impact of AI infrastructure on land and people.

math.HO↗

arXiv2026

Subgraph Entropy

Given $r \geq 3$, we prove that there exists $λ>0$ depending only on $r$ so that if $G$ is a metric graph of rank $r$ with metric entropy $1$, then there exists a proper subgraph $H$ of $G$ with metric entropy at least $λ$. This answers a question of the second two authors together with Rieck. We interpret this as a graph theoretic version of the Bers Lemma from hyperbolic geometry, and explain some connections to the pressure metric on the Culler-Vogtmann Outer Space.

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arXiv2026

On fixed points of pseudo-Anosov maps

We give a formula to estimate the number of fixed points of a pseudo-Anosov homeomorphism of a surface. When the homeomorphism satisfies a mild property called strong irreducibility, the log of the number of fixed points is coarsely equal to the Teichmuller translation length. We also discuss several applications, including an inequality relating the hyperbolic volume of a mapping torus to the rank of its Heegaard Floer homology.

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arXiv2026

On the monodromy and spin parity of single-cylinder origamis in the minimal stratum

In a paper with Menasco-Nieland, the first author constructed factorially many origamis in the minimal stratum of the moduli space of translation surfaces having simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these origamis were constructed using the minimal number of squares required for origamis in the minimal stratum. We shall call such origamis minimal $[1,1]$-origamis. In this work, we calculate all of the spin parities of the Aougab-Menasco-Nieland origamis, and we therefore determine the connected component of the minimal stratum within which each is contained. Motivated by understanding the $\SL(2,\Z)$-orbits of these origamis, we investigate their monodromy groups, in particular proving that all of them are alternating or projective special linear groups. In fact, we prove more generally that the monodromy group of a minimal $[1,1]$-origami must almost always be a finite simple group. Finally, we determine the Kontsevich-Zorich monodromies of these origamis in low genus and give a conjecture in general. Note that previous works in the literature (e.g., that of Eskin-Kontsevich-Zorich, Filip-Forni-Matheus, Gutiérrez-Romo, Kany-Matheus, Matheus-Yoccoz-Zmiaikou, and Zorich) often chose to discuss just one of these $\SL(2,\Z)$-invariants at a time: in particular, to the best our knowledge, this is one of the first places where all of these $\SL(2,\Z)$-invariants are computed explicitly in a single paper for such a large family of origamis.

math.GT↗

arXiv2026

Quasi-isometric rigidity of the integers: an elementary primer

Chatawate (Flame) Ruethaimetapat was a passionate, enthusiastic, and wonderful person who passed away in August of 2024. At the time of their passing they were working towards their PhD, specializing in geometric group theory. Flame was just as excited about learning new mathematics as they were about sharing it with everyone else, so it's no surprise that they spent a lot of time thinking about how to write down expository proofs of classical theorems that would be accessible for first year students. In particular, they sought a simple, elementary proof of the fact that any finitely generated group quasi-isometric to the integers is virtually the integers. In the spirit of this endeavor and in loving memory of Flame, we present such a proof here.

math.GR↗

arXiv2025

Pseudo-Anosovs from the perspective of their mapping tori

In this chapter, we outline some of the many combinatorial tools developed over the past three decades for studying a pseudo-Anosov diffeomorphism of a surface by analyzing the geometry of its mapping torus. We begin with an overview of the various simplicial complexes associated with a surface (such as the curve, arc, and pants complexes) and explain how to relate the dynamics of the action of a given pseudo-Anosov on any one of these complexes to the dynamics of the diffeomorphism itself, or to the hyperbolic geometry of its mapping torus. We next cover some of the more modern features of the theory by discussing various analogs of pseudo-Anosov diffeomorphisms on surfaces of infinite type. We conclude with a description of original work-- due jointly to the author with Dave Futer and Sam Taylor-- that relates the action of a pseudo-Anosov on the curve complex to the minimum number of fixed points for any map in the corresponding isotopy class. The paper is written in as accessible a way as possible while assuming only the bare minimum in background. The hope is to informally convey to the reader some of the main ideas and strategies in the area.

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arXiv2025

Constructing reducibly geometrically finite subgroups of the mapping class group

In this article, we consider qualified notions of geometric finiteness in mapping class groups called parabolically geometrically finite (PGF) and reducibly geometrically finite (RGF). We examine several constructions of subgroups and determine when they produce a PGF or RGF subgroup. These results provide a variety of new examples of PGF and RGF subgroups. Firstly, we consider the right-angled Artin subgroups constructed by Koberda and Clay--Leininger--Mangahas, which are generated by high powers of given elements of the mapping class group. We give conditions on the supports of these elements that imply the resulting right-angled Artin subgroup is RGF. Secondly, we prove combination theorems which provide conditions for when a collection of reducible subgroups, or sufficiently deep finite-index subgroups thereof, generate an RGF subgroup.

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arXiv2025

The arc complex is not quasi-isometric to the sphere complex

We show that the arc complex $\mathcal{A}(S_{g,1})$ is not quasi-isometric to the sphere complex $\mathcal S_{2g}$ associated to the double of a genus $2g$ handlebody. Along the way, we present a simple proof that $\mathcal{A}(S_{g,1})$ is quasi-isometrically rigid.

math.GT↗

arXiv2023

Currents with corners and counting weighted triangulations

Let $Σ$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $Σ$, which behaves like a geodesic current away from certain singularities (the "corners"). We topologize the space of all currents with corners and study its properties. We prove that the space of currents with corners shares many properties with the space of geodesic currents, although crucially, there is no canonical action of the mapping class group nor is there a continuous intersection form. To circumvent these difficulties, we focus on those currents with corners arising from harmonic maps of graphs into $Σ$. This leads to the space of \textit{marked harmonic currents with corners}, which admits a natural Borel action by the mapping class group, and an analog of Bonahon's\cite{Bonahon} compactness criterion for sub-level sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on $Σ$ for triangulations. Fixing an embedding $ϕ$ of a weighted graph $Γ$ into $Σ$ whose image $ϕ(Γ)$ is a triangulation of $Σ$, let $N_ϕ(L)$ denote the number of mapping classes $f$ so that a weighted-length minimizing representative in the homotopy class determined by $f \circ ϕ$ has length at most $L$. In analogy with theorems of Mirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, and Rafi-Souto\cite{RafiSouto}, we prove that $N_ϕ(L)$ grows polynomially of degree $6g-6$ and the limit \[ \lim_{L \rightarrow \infty} \frac{N_ϕ(L)}{L^{6g-6}}\] exists and has an explicit interpretation depending on the geometry of $Σ$, the vector of weights, and the combinatorics of $ϕ$ and $Γ$.

math.GT↗

arXiv2023

Unmarked simple length spectral rigidity for covers

We prove that every closed orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the sets of lengths associated to curves with at most k self-intersections differ.

math.GT↗

arXiv2023

A note on an effective characterization of covers with an application to higher rank representations

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the hypotheses to consider curves with explicitly bounded self-intersection number. As an application we show that for sufficiently large N, the set of unmarked traces associated to simple closed curves in a generically chosen representation to SL(N, R) distinguishes between pairs of non-isomorphic covers.

math.GT↗

arXiv2023

Automorphisms of the k-curve graph

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.

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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 $γ_{n}$ 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 Teichmüller space at which $γ_{n}$ is length-minimized stays in some compact set; (2) the self-intersection number of $γ_{n}$ is on the order of $n^{2}$, the minimum length of $γ_{n}$ taken over all hyperbolic metrics is on the order of $n$, and the metric minimizing the length of $γ_{n}$ 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 $γ_{n}$ admits a simple elevation (which we call the $\textit{simple lifting degree}$ of $γ_{n}$) 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.

math.GT↗

arXiv2022

Covers of surfaces, Kleinian groups, and the curve complex

We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite covers. As applications, we effectively relate the electric circumference of a fibered manifold to the curve complex translation length of its monodromy, and we give quantitative bounds on virtual specialness for cube complexes dual to curves on surfaces.

math.GT↗

arXiv2022

Origamis associated to minimally intersecting filling pairs

Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single topological disk. Using different techniques, we improve on this result by constructing factorially-many (again in $g$) such orbits. These new orbits are chosen so that the absolute value of the algebraic intersection number is equal to the geometric intersection number, implying that each pair naturally gives rise to an origami. We collect some rudimentary experimental data on the corresponding $SL(2, \mathbb{Z})$-orbits and suggest further study and conjectures.

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arXiv2021

Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

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arXiv2020

Thermodynamic metrics on outer space

In this paper we consider two piecewise Riemannian metrics defined on the Culler-Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil-Petersson metric on the Teichmüller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil-Petersson metric, from the point of view of geometric group theory, these metrics behave very differently to the Weil-Petersson metric. Specifically, we show that when the rank $r$ is at least 4, the action of ${\rm Out}(\mathbb{F}_r)$ on the completion of the Culler-Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.

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