Related papers: Vertex-minimal hyperbolic origami 2-torus

Let M be an orientable hyperbolic surface without boundary and let $\gamma$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $\gamma$ in H2 is shorter than $\gamma$.

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped…

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ have 11 and 12 vertices,…

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable…

Let $M$ be an orientable 3-manifold with $\partial M$ a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most $g$ is bounded by a quadratic function of $g$. In the hyperbolic case, this…

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.

We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.

In the early 2000s, Frigerio, Martelli, and Petronio studied $3$-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class $M_{g,k}$ of smallest complexity…

For $n \geq 15$, we prove that the minimum number of triangles in an $n$-vertex $K_4$-saturated graph with minimum degree 4 is exactly $2n-4$, and that there is a unique extremal graph. This is a triangle version of a result of Alon,…

The string vertices of closed string field theory are subsets of the moduli spaces of punctured Riemann surfaces that satisfy a geometric version of the Batalin-Vilkovisky master equation. We present a homological proof of existence of…

This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of…

It is shown that each quadrangulation of the 2-torus by the Cartesian product of two cycles can be geometrically realized in (Euclidean) 4-space without hidden symmetries---that is, so that each combinatorial cellular automorphism of the…

Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point…

Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion $M_i$ of $M$ and modify the…

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks…

The Bratteli diagram is an infinite graph which reflects the structure of projections in a C*-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination with the…

Let $\mathbb{M}^{2}$ be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in $\mathbb{M}^{2}\times\mathbb{R}$. First, using a…

We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$…

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is…

We show that the algebraic intersection form of hyperbolic surfaces of genus $g$ has a minimum in the moduli space and that the minimum grows in the order $(\log g)^{-2}$ in terms of the genus. We also describe the asymptotic behavior of…