arXiv research

Lower bounds for Hodge-Laplacian spectrum on orbifolds.

Study essential spectrum of differential operators on geometrically finite orbifolds.

Study bottom of spectra on orbifolds via coverings.

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collecti…

The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is …

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result …

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…

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.

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product …

In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is ho…

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…

Researchers prove finiteness of integral representations on specific polytopes.

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…

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…

We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, p…

We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry…

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 …

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

Study topological quantum mechanics on orbifolds with geometric interpretation.

Study geometrically measures to decide if modular companions are conformally equivalent.

This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.

The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.

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

Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical sys…

The paper classifies fibrations of 3-dimensional flat orbifolds.

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…

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…

We prove that a connected 2-dimensional orbifold with finitely generated and infinite orbifold fundamental group is good. We also describe all the good 2-dimensional orbifolds with finite orbifold fundamental groups

A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman along with earlier work of Boileau-Leeb-Porti and Cooper-Hodgson-Kerckhoff. We give a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow.…

We propose a dimensional reduction procedure in the Stolz--Teichner framework of supersymmetric Euclidean field theories (EFTs) that is well-suited in the presence of a finite gauge group or, more generally, for field theories over an orbifold. As an illustration, we give a geometric interpretation of the Chern charact…

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.

In this article we prove a generalization of Selberg's lemma on the existence of torsion free, finite index subgroups of arithmetic groups. Some of the geometric applications are the resolution a conjecture of Nimershiem and answers to questions of Long-Reid and the author.

A Calabi-Yau orbifold is locally modeled on C^n/G where G is a finite subgroup of SL(n, C). In dimension n=3 a crepant resolution is given by Nakamura's G-Hilbert scheme. This crepant resolution has a description as a GIT/symplectic quotient. We use tools from global analysis to give a geometrical generalization of the…

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 introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for equivariant cohomotopy which works for any compact Lie group G. In the special case when G is finite or a torus, we show that ou…

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Ge…

Study of orbifold mapping class groups via arc and curve actions.

We determine the extent to which the collection of $Γ$-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of $Γ$-Euler-Satake characteristics corresponding to free or free abelian $Γ$ and are not…

We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between …

We prove that Demailly's holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric criterion for a compact connected orbifold to be a Moishezon orbifolds, thus generalizin…

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

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 construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …

Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly …

Unified theory of orbifolds and cohomology.

We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem for branchfolds and apply it to prove that a conifold can be endowed with branc…

The study constructs a hyperbolic orbifold to show how certain Salem numbers can be realized geometrically.