Do and Norbury found a so-called differential relation which relates the volume of the moduli space of singular surface with a cone point to that of a smooth surface obtained by forgetting the cone point. Their procedure is valid for cone angles less than $π$ by work of Tan, Wong and Zhang. We study the moduli space of a surface with a single cone point of angle ranging from $0$ to $2π$ using a coordinate system closely related to Penner's $λ$-lengths. We compute the action of the mapping class group in these coordinates, give an explicit expression for Wolpert's symplectic form and use this to justify Do and Norbury's approach using just hyperbolic geometry.
We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.
In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the 3-sphere.
We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any $n\geq4$, we construct a finite subgraph $X_n$ of the pants graph $P(S_{0,n})$ of the n-punctured sphere $S_{0,n}$ with the following property. Any simplicial embedding of $X_n$ into any pants graph $P(S_{0,m})$ of a punctured sphere is induced by an embedding $S_{0,n} \to S_{0,m}$.
Let $S_{0,n}$ be an $n$-punctured sphere. For $n\geq 4$, we construct a sequence $(\mathcal{X}_i)_{i\in\mathbb{N}}$ of finite rigid sets in the pants graph $\mathcal{P}(S_{0,n})$ such that $\mathcal{X}_1 \subset \mathcal{X}_2 \subset ...\subset\mathcal{P}(S_{0,n})$ and $\bigcup_{i\geq1}\mathcal{X}_i=\mathcal{P}(S_{0,n})$.
We classify the topological types for the unions of the totally geodesic 3-punctured spheres in orientable hyperbolic 3-manifolds. General types of the unions appear in various hyperbolic 3-manifolds. Each of the special types of the unions appears only in a single hyperbolic 3-manifold or Dehn fillings of a single hyperbolic 3-manifold. Furthermore, we investigate bounds of the moduli of adjacent cusps for the union of linearly placed 3-punctured spheres.
Every closed hyperbolic geodesic $γ$ on the triply--punctured sphere $M =\widehat{\mathbb C} - \{0,1,\infty\}$ has a self--intersection number $I(γ) \ge 1$ and a combinatorial length $L(γ) \ge 2$, the latter defined by the number of times $γ$ passes through the upper halfplane. In this paper we show that $δ(γ) = I(γ) - L(γ) \ge -1$ for all closed geodesics; and that for each fixed $δ$, the number of geodesics with invariants $(δ,L)$ is given exactly by a quadratic polynomial $p_δ(L)$ for all $L \ge 4 + δ$.
A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.
Let $Γ$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $Γ$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing rela…
We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to describe the minimum of the spine systole on the moduli space of punctured surfaces.
In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + αD$ for $α\in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.
We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for $n=7,10,11$, arithmetic surfaces do not achieve the maximal systole length.
We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and F_D-type Lauricella hypergeometric functions.
We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.
We exhibit the traceless $SU(2)$ character variety of a 6-punctured 2-sphere as a 2-fold branched cover of ${\mathbb{C}}P^3$, branched over the singular Kummer surface, with the branch locus in $R(S^2,6)$ corresponding to the binary dihedral representations. This follows from an analysis of the map induced on $SU(2)$ character varieties by the 2-fold branched cover $F_{n-1}\to S^2$ branched over $2n$ points, combined with the theorem of Narasimhan-Ramanan which identifies $R(F_2)$ with ${\mathbb{C}} P^3$. The singular points of $R(S^2,6)$ correspond to abelian representations, and we prove tha…
For an appropriate choice of a $\mathbb{Z}$-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a $(k+3)$-punctured sphere, i.e. the Weinstein manifold given as the complement of $(k+3)$ generic lines in $\mathbb{C}P^2$ is quasi-equivalent to the derived category of coherent sheaves on a singular surface $\mathcal{Z}_{2,k}$ constructed as the boundary of a toric Landau-Ginzburg model $(\mathcal{X}_{2,k}, \mathbf{w}_{2,k})$. We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localising. We also …
We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.
Deligne and Mostow constructed a class of lattices in PU(2,1) using monodromy of hypergeometric functions. Later, Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for all lattices with three fold symmetry in Deligne-Mostow list. This is a generalisation of the works of Parker and Boadi and gives a different interpretation of the fundamental domain constructed by Deraux, Falbel and Paupert for some of these lattices.
We study the space $C(a_0,a_1,\dots,a_n)$ of hyperbolic 2-spheres with cone points of prescribed apex curvatures $2a_0,2a_1,\dots,2a_n\in]0,2π[$ and some related spaces. For $n=3$, we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for $n=4$, the corresponding spaces provide the famous 7 examples of nonarithmetic compact holomorphic 2-ball quotients previously constructed by Deligne-Mostow.
In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of maps which differ by exponentially decaying variations. Previously, harmonic maps from once-punctured spheres to the hyperbolic plane have been parameterized by holomorphic quadratic differentials on the complex plane. Our harmonic maps, mapping a genus g>1 domain to a k-sided polygon, correspond to meromorphic quadratic differentials having one pole of order (…
For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures on closed surfaces, and settles a long-standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock-Goncharov coordinates of t…
We construct embeddings of simplicial complexes into a (surface of a) simplicial ball whose triangulation has bounded degrees and low volume. This construction can be used either to efficiently "simplify a complicated space" by realizing it as a part of a ball/sphere, or to "complexify" a sphere - to give it a specific metric that inherits desired properties from an embedded complex.
