Searcharxiv⌕ Search

arXiv subjects

Tarik Aougab

Publications and source records attributed to Tarik Aougab.

At least 37 records · Page 2Linked to original sources
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.

math.GT↗
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.

math.GT↗
arXiv2020

Characterizing covers via simple closed curves

Given two finite covers $p: X \to S$ and $q: Y \to S$ of a connected, oriented, closed surface $S$ of genus at least $2$, we attempt to characterize the equivalence of $p$ and $q$ in terms of which curves lift to simple curves. Using Teichmüller theory and the complex of curves, we show that two regular covers $p$ and $q$ are equivalent if for any closed curve $γ\subset S$, $γ$ lifts to a simple closed curve on $X$ if and only if it does to $Y$. When the covers are abelian, we also give a characterization of equivalence in terms of which powers of simple closed curves lift to closed curves.

math.GT↗
arXiv2017

Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.

math.GT↗
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.

math.GT↗
arXiv2016

Counting curve types

Let $S$ be a closed orientable hyperbolic surface, and let $\mathcal{O}(K,S)$ denote the number of mapping class group orbits of curves on $S$ with at most $K$ self-intersections. Building on work of Sapir [16], we give upper and lower bounds for $\mathcal{O}(K,S)$ which are both exponential in $\sqrt{K}$.

math.GT↗
arXiv2016

Building hyperbolic metrics suited to closed curves and applications to lifting simply

Let $γ$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $ρ$ for which $γ$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreover, the injectivity radius of $ρ$ is at least $1/(2\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which $γ$ lifts as a simple closed curve (i.e. lifts simply). We also show that if $γ$ is a closed curve with length at most $L$ on a cusped hyperbolic surface $\mathcal{S}$, then there exists a cover of $\mathcal{S}$ of degree at most $N \cdot L \cdot e^{L/2}$ to which $γ$ lifts simply, for $N$ depending only on the topology of $\mathcal{S}$.

math.GT↗
arXiv2016

Constructing large k-systems on Surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malnič-Mohar \cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_{g}$ whose size is on the order of $g^{k/4}$. For each $k\geq 2$, We construct a $k$-system on $S_{g}$ with on the order of $g^{\lfloor (k+1)/2 \rfloor +1}$ elements. The $k$-systems we construct behave well with respect to subsurface inclusion, analogously to how a pants decomposition contains pants decompositions of lower complexity subsurfaces.

math.GT↗
arXiv2015

Local geometry of the k-curve graph

Let $S$ be an orientable surface with negative Euler characteristic. For $k \in \mathbb{N}$, let $\mathcal{C}_{k}(S)$ denote the $\textit{k-curve graph}$, whose vertices are isotopy classes of essential simple closed curves on $S$, and whose edges correspond to pairs of curves that can be realized to intersect at most $k$ times. The theme of this paper is that the geometry of Teichmüller space and of the mapping class group captures local combinatorial properties of $\mathcal{C}_{k}(S)$. Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large $k$: the clique number of $\mathcal{C}_{k}(S)$ (exponential in $k$, which improves on all previously known bounds and which is essentially sharp); the maximum size of the intersection, whenever it is finite, of a pair of links in $\mathcal{C}_{k}$ (quasi-polynomial in $k$); and the diameter in $\mathcal{C}_{0}(S)$ of a large clique in $\mathcal{C}_{k}(S)$ (uniformly bounded). As an application, we obtain quasi-polynomial upper bounds, depending only on the topology of $S$, on the number of short simple closed geodesics on any square-tiled surface homeomorphic to $S$.

math.GT↗
arXiv2015

Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length

Given $ϕ$ a pseudo-Anosov map, let $\ell_\mathcal{T}(ϕ)$ denote the translation length of $ϕ$ in the Teichmüller space, and let $\ell_\mathcal{C}(ϕ)$ denote the stable translation length of $ϕ$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $χ(S)$, the minimal possible ratio $τ(ϕ) = \frac{\ell_\mathcal{T}(ϕ)}{\ell_\mathcal{C}(ϕ)}$ is $\log(|χ(S)|)$, up to uniform additive and multiplicative constants. In this short note, we introduce a new construction of such ratio optimizers and demonstrate their abundance in the mapping class group. Further, we show that ratio optimizers can be found arbitrarily deep into the Johnson filtration as well as in the point pushing subgroup.

math.GT↗
arXiv2015

Curves intersecting exactly once and their dual cube complexes

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the first author and further answering a question of Malestein-Rivin-Theran. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev. In particular, we show that for any even $k$ between $\lfloor g/2 \rfloor$ and $g$, there exists such collections whose dual cube complexes have dimension $k$, and we prove a simplifying structural theorem for any cube complex dual to a collection of curves on a surface pairwise intersecting at most once.

math.GT↗
arXiv2013

Minimally intersecting filling pairs on surfaces

Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function $\mathcal{F}_{g}$ over the Moduli space of Riemann surfaces of genus $g$, which, given a hyperbolic metric $σ$, outputs the length of the shortest, minimally intersecting filling pair for the metric $σ$. We completely characterize the global minima of $\mathcal{F}_{g}$, and using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grow at least exponentially in $g$.

math.GT↗
arXiv2013

On curves and polygons with the equiangular chord property

Let $C$ be a smooth, convex curve on either the sphere $\mathbb{S}^{2}$, the hyperbolic plane $\mathbb{H}^{2}$ or the Euclidean plane $\mathbb{E}^{2}$, with the following property: there exists $α$, and parameterizations $x(t), y(t)$ of $C$ such that for each $t$, the angle between the chord connecting $x(t)$ to $y(t)$ and $C$ is $α$ at both ends. Assuming that $C$ is not a circle, E. Gutkin completely characterized the angles $α$ for which such a curve exists in the Euclidean case. We study the infinitesimal version of this problem in the context of the other two constant curvature geometries, and in particular we provide a complete characterization of the angles $α$ for which there exists a non-trivial infinitesimal deformation of a circle through such curves with corresponding angle $α$. We also consider a discrete version of this property for Euclidean polygons, and in this case we give a complete description of all non-trivial solutions.

math.DG↗
arXiv2013

Small intersection numbers in the curve graph

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p \ge 0$ punctures, and let $ω(g,p)= 3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ in the curve graph satisfying the following optimal intersection property: for any natural number $k$, the endpoints $v_{i},v_{i+k}$ of any length $k$ subsegment intersect $O(ω^{k-2})$ times. By combining this with work of the first author, we answer a question of Dan Margalit.

math.GT↗
arXiv2013

Quadratic Bounds on the Quasiconvexity of Nested Train Track Sequences

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitting and sliding sequences of birecurrent train tracks project to $O((g,p)^{2})$-unparameterized quasi-geodesics in the curve graph of any essential subsurface, an effective version of a theorem of Masur, Mosher, and Schleimer.

math.GT↗
arXiv2012

Uniform Hyperbolicity of the Graphs of Curves

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $δ$-hyperbolic for some $δ=δ(g,p)$. In this paper, we show that there exists some $δ>0$ independent of $g,p$ such that the curve graph $\mathcal{C}_{1}(S_{g,p})$ is $δ$-hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with $g$ and $p$: the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichmüller space to $\mathcal{C}(S)$ sending a Riemann surface to the curve(s) of shortest extremal length.

math.GT↗
arXiv2012

Large Collections of Curves Pairwise Intersecting Exactly Once

Let $Ω=(ω_{j})_{j\in I}$ be a collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus $g$ surface $S_{g}$, such that $ω_{i}$ and $ω_{j}$ intersect exactly once for $i\neq j$. It was recently demonstrated by Malestein, Rivin, and Theran that the cardinality of such a collection is no more than $2g+1$. In this paper, we show that for $g\geq 3$, there exists at least two such collections with this maximum size up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran.

math.GT↗