Research

Abstract. 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.

Abstract. 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>=p>=0. 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.

Abstract. 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 an identity for the sum of Lagrange numbers.

Abstract. 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 Teichmüller 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 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 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.

Abstract. 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(√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(k21/40), and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain O(√k log k). By contrast, our methods are elementary, combinatorial, and geometric. 

Abstract. 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. 

Abstract. A collection of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number is equal to k in absolute value for each pair in the collection. Generalizing a theorem of Malestein-Rivin-Theran, we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g+1 when g ≥ 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.

Abstract. Przytycki has shown that the size Nk(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 k2 + k + 1. In this paper, we narrow Przytycki’s bounds by showing that Nk(S) is O(|χ(S)|^{3k} / (log|χ(S)|)^2).

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 Nk(S) ≤ O(n2k+2).

Using similar techniques, we also obtain the sharp estimate N2(S) = Θ(n3) when k = 2 and g is fixed. 

Abstract. Let Sg 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 Sg which pairwise intersect exactly once, extending a result of the first author [1] and further answering a question of Malestein-Rivin-Theran [10]. To distinguish such collections up to the action of the mapping class group, we analyze their dual cube complexes in the sense of Sageev [12]. In particular, we show that for any even k between g/2 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. 

Abstract. Let γ be an essential closed curve with at most k self-intersections on a surface S with negative Euler characteristic. In this paper, we construct a hyperbolic metric ρ for which γ has length at most M ·√k, where M is a constant depending only on the topology of S. Moreover, the injectivity radius of ρ is at least 1/(2 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 S, then there exists a cover of S of degree at most N ·L·eL/2 to which γ lifts simply, for N depending only on the topology of S. 

Abstract. In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichmuller space that depend on the dual cube complex associated to the collection, a concept due to Sageev. As an application of our bounds, we obtain estimates for the ‘longest’ curve with k self-intersections, complementing work of Basmajian [Bas]. 

Abstract. Let fρ(L) indicate the smallest integer so that every curve on a fixed hyperbolic surface (S,ρ) of length at most L lifts to a simple curve on a cover of degree at most fρ(L). We provide linear lower bounds for fρ(L), improving a recent result of Gupta-Kapovich [6]. When (S, ρ) is without punctures, using work of Patel [9] and Lenzhen-Rafi-Tao [7] we conclude that fρ(L)/L grows like the reciprocal of the systole of (S,ρ). When (S,ρ) has a puncture, we obtain lower bounds for fρ that are exponential in L. 

Abstract. 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. 

Abstract. In our previous paper [GLM18], we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights. We also present a systematic method to construct such weighted triangulations in the 2-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane. 

Abstract. 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. 

Abstract. 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. 

Here is a webpage with more detail about Harmony; also check it out on github.

Abstract. With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze two natural variants of the Farey graph F in which we relax the edge condition to indicate intersection number = k or ≤ k, respectively. 

The former 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 is a quasi-tree (in fact, a tree when k is even) and its automorphism group is uncountable for k > 1. 

As for intersection number ≤ k, Agol obtained an upper bound of 1 + min{p : p 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 (mod 12), and we show computer-assisted computations of many values of k for which equality fails. 

Abstract. We prove that there is an algorithm to determine if a given finite graph is an induced subgraph of a given curve graph. 

Abstract. We say a graph has property Pg,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 Pg,p. We introduce a new invariant of a graph, the nested complexity length, which provides a novel obstruction to Pg,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 Pg,p, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have Pg,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 complete multipartite graphs cannot have Pg,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 Pg,p. 

Abstract. 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. 

Abstract. 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 G. 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”. 

Abstract. Certain classes of 3-manifolds, following Thurston, give rise to a ‘skinning map’, a self-map of the Teichmuller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map associated to M sends a specified path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M produces examples of non-immersion skinning maps on the Teichmuller spaces of surfaces in each even genus, and with either 4 or 6 punctures.