arXiv research

The paper studies orbifold splice quotients and log covers of surface pairs.

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They were also able to show that under an additional hypothesis on M called the congr…

Existence of Kähler-Einstein metrics on toric varieties proven.

To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their unive…

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negativ…

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let $G$ be a Lie group acting on a space $X$. We show that the space of isotopy-equivalence classes of $(G,X)$-structures …

New theorem extends Hadamard's to transversely affine geometry.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

Study bottom of spectra on orbifolds via coverings.

New orbifolds found that cover manifolds but not in a finite way.

We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.

We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …

We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.

In this article we examine the conjecture of Neumann and Reid that the only hyperbolic knots in the $3$-sphere which admit hidden symmetries are the figure-eight knot and the two dodecahedral knots. Knots whose complements cover hyperbolic reflection orbifolds admit hidden symmetries, and we verify the Neumann-Reid con…

The paper generalizes Nielsen equivalence to 2-orbifolds.

We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or ho…

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti…

Hyperbolic knots decompose into prism orbifolds.

The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex …

A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metric…

Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold Sigma whose fundamental group is Gamma. Our orbifold invariants have a natural and…

For any $n>1$, we construct examples branched Galois coverings from $M$ to the nth projective space ${\mathbb P}^n$ where $M$ is one of $({\mathbb P}^1)^n$, ${\mathbb C}^n$ or $(B_1)^n$, and $B_1$ is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over ${\mathbb P}^n$ uniformized by $M$.…

We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.

Computes a finite presentation for P(SL(2,Z)) from spin mapping class group.

This work concludes a series of four papers on the foundational theory of orbifolds and stacks. We apply the abstract theory, developed in its predecessors, to orbifolds derived from manifolds. Specifically, we show how the very concrete topological base spaces associated to such orbifolds can be described and manipula…

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determin…

We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some $k\geq 2$. We also obtain a topological classification in…

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the numbe…

We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential Topology/Geometry, including orbibundles, differential forms, integration and (equivariant) …

Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smooth…

For a finitely generated discrete group $Γ$, the $Γ$-sectors of an orbifold $Q$ are a disjoint union of orbifolds corresponding to homomorphisms from $Γ$ into a groupoid presenting $Q$. Here, we show that the inertia orbifold and $k$-multi-sectors are special cases of the $Γ$-sectors, and that the $Γ$-sectors are orbif…

We show that the "geometric models of matter" approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through…

The paper classifies cusp types in hyperbolic knot complements.

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a char…

The study of universal links in 3-manifolds and their properties.

These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties thro…

We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber …

In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the or…

We build a concrete and natural model for the strict 2-category of orbifolds. In particular we prove that if one localizes the 2-category of proper etale Lie groupoids at a class of 1-arrows that we call "covers", then the strict 2-category structure drops down to the localization. In our construction the spaces of 1- …

We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the "turnover core,'' whose volume is bounded from above by a function…

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.

We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either by puncturing a closed surface or via orbifold coverings. As a corollary, we give new quasi-isometric embeddings between mapping class groups.

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime…

New method shows any triangle group generating pair is related to special coverings.

The paper explores simple and relatively simple transformation groups and their universal coverings.

Compact Kähler manifolds with positive curvature have contractible covers.