arXiv research

In early 1930s Seifert and Threlfall classified up to conjugacy the finite subgroups of $\mathrm{SO}(4)$, this gives an algebraic classification of orientable spherical 3-orbifolds. For the most part, spherical 3-orbifolds are Seifert fibered. The underlying topological space and singular set of non-fibered spherical 3…

We extend Matveev's theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible 3-orbifolds, and, with certain restrictions and subtleties, additivity under orbifold connected sum. We also develop the theory of…

The abstract constructs a set of bad 3-orbifolds and shows how any bad 3-orbifold can be transformed into a good one.

Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.

Study spectral gaps and bass notes of random hyperbolic 3-orbifolds.

Study projective deformations of hyperbolic 3-orbifolds with turnover ends.

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.

We prove every oriented compact cyclic $3$-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compa…

Researchers found a new hyperbolic 3-orbifold using a Menger curve.

We study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. We show that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds either admits a metric of nonnegative sectional curvature or satisfies Thurston's Geometrization Conjectur…

We construct the first example of a ``one-cusped'' hyperbolic 3-orbifold for which we see the true shape of the space of hyperbolic Dehn fillings.

We study the isometry groups of compact spherical orientable $3$-orbifolds $S^3/G$, where $G$ is a finite subgroup of $\mathrm{SO}(4)$, by determining their isomorphism type. Moreover, we prove that the inclusion of $\mbox{Isom}(S^3/G)$ into $\mbox{Diff}(S^3/G)$ induces an isomorphism of the $π_0$ groups, thus proving …

We give examples of open 3-manifolds and 3-orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2-suborbifolds) with nonnegative Euler characteristic.

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…

Neumann and Reid described in their paper "Rigidity of cusps in deformations of hyperbolic 3-orbifolds" (Math Ann. 295 (1993) no. 2, 223--237) a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' snappea program, they conjectured that the cu…

We establish a lower bound on the complexity orientable locally orientable geometric 3-orbifolds in terms of Delzant's T-invariants of their orbifold-fundamental groups, generalizing previously known bounds for complexity of 3-manifolds.

We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the …

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

We prove the following result: Let $(\mathcal{O},g_0)$ be a complete, connected 3-orbifold with uniformly positive scalar curvature, with bounded geometry, and containing no bad 2-suborbifolds. Then there is a finite collection $\mathcal{F}$ of spherical 3-orbifolds, such that $\mathcal{O}$ is diffeomorphic to a (possi…

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…

Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.

A formula is given which computes the Seiberg-Witten invariant of a 3-orbifold from the invariant of the underlying manifold. As an application, we derive a formula for the Seiberg-Witten invariant of a non-Kähler complex surface, which was originally due to O. Biquard \cite{Biq} and S.R. Williams \cite{W} independentl…

Researchers solved the multiple fibration problem for Seifert 3-orbifolds.

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…

Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.

For each natural number n >= 4, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by n. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked n. We apply this result to give sharp lower bounds on the volume of a hyperbolic m…

Using bordered Floer theory, we construct an invariant $\widehat{\mathit{HFO}}(Y^{\text{orb}})$ for $3$-orbifolds $Y^{\text{orb}}$ with singular set a knot that generalizes the hat flavor $\widehat{\mathit{HF}}(Y)$ of Heegaard Floer homology for closed $3$-manifolds $Y$. We show that for a large class of $3$-orbifolds,…

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques involve computing the guts of the orbifold split along the $2$-suborbifold via a …

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of total…

The paper classifies fibrations of 3-dimensional flat orbifolds.

The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.

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…

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds wit…

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently construct…

New method shows how certain groups act on 3-orbifolds.

We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.

The famous Haken-Kneser-Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds wo…

Paper distinguishes 2-knots with circle actions using fundamental groups.

We prove that a closed 3-orbifold that fibers over a hyperbolic polygonal 2-orbifold admits a family of hyperbolic cone structures that are viewed as regeneration of the polygon, provided that the perimeter is minimal.

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold defines $c Q^{1/2} + O(Q^{1/4})$ square-rootable Salem numbers of degree $4$ which are…

We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) cover…

Develops new methods for isospectral orbifolds and regulator quotients.

New Heegaard Floer homology for orbifolds with cyclic singularities.

This paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the 3-sphere, or more generally in a Z_6 homology sphere. We also prove more general lower bounds under mild homological hypot…

Odd Fibonacci groups cannot form hyperbolic 3-orbifolds.

We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental group. The first proof is direct and topological; the second is group-theoretic. Th…

The purpose of this article is to give a proof of the Orbifold Theorem announced by Thurston in late 1981: If $O$ is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then $O$ is geometric. As a corollary, any smooth orientation preserving non-free f…