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 on a compact orbifold $Σ$ is locally homeomorphic to the space of representations of the orbifold fundamental group of $Σ$ to $G$ following the work of Thurston, Morgan, and Lok. This implies that the deformation space of $(G, X)$-structures on $Σ$ is locally homeomorphic to the space of representations of the orbifold fundamental group to $G$ when restricted to the region of proper conjugation action by $G$.
In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan's theory of twisted orbifold cohomology and the local system of Freed-Hopkins-Teleman in their work in twisted K-theory. In the case in which the orbifold is simply a manifold we recover previous results of Gawedzki and Brylinski.
Motivated by Kohno's result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice and a free Lie subalgebra. This yields the structure of the holonomy Lie algebra of a finite hypersolvable (including supersolvable) lattice. As applications, we obtain the structure of the holonomy Lie algebra of (the Salvetti complex of) a supersolvable oriented matroid, and tha…
Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of properly convex ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. For this, we will study the natural conditions on eigenvalues of holonomy r…
We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including t…
Let $\mathcal{O}$ be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form $ω$ on the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$. We show that the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$ admits a global Darboux coordinates system with respect to $ω$. To this end, we show that $\mathcal{C}(\mathcal{O})$ can be decomposed into smaller symplectic spaces. In the course of the proof, we also study the deformation spa…
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.
Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study …
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 cusps are strongly geometrically isolated, a fact which we establish here. We also give a parameterization of the Dehn surgery space of this orbifold which has amusing properties.
Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural condit…
In this paper we investigate the properties of the real and complex projective structures associated to Hitchin and quasi-Hitchin representations that were originally constructed using Guichard-Wienhard's theory of domains of discontinuity. We determine the topology of the underlying manifolds and we prove that some of these geometric structures are fibered in a special standard way. In order to prove these results, we give two new ways to construct these geometric structures: we construct them using gauge theory, flat bundles and Higgs bundles, and we also give a new geometric way to construc…
We conclude the multiple fibration problem for closed orientable Seifert three-orbifolds, namely the determination of all the inequivalent fibrations that such an orbifold may admit. We treat here geometric orbifolds with geometries $\mathbb R^3$ and $\mathbb S^2\times\mathbb R$ and bad orbifolds (hence non-geometric), since the only other geometry for which the multiple fibration phenomenon occurs, namely $\mathbb S^3$, has been treated before by the second and third author. For the geometry $\mathbb R^3$ we recover, by direct and geometric arguments, the computer-assisted results obtained by…
In this paper, we develop the theory for classifying all the geometric fibrations of compact, connected, flat $n$-orbifolds, over a 1-orbifold, up to affine equivalence. We apply our classification theory to classify all the geometric fibrations of compact, connected, flat $2$-orbifolds, over a 1-orbifold, up to affine equivalence. This paper is an essential part of our project to give a geometric proof of the classification of all closed flat 4-manifolds.
This paper is a continuation of our previous paper, Co-Seifert fibrations of compact flat orbifolds, in which we developed the theory for classifying geometric fibrations of compact, connected, flat $n$-orbifolds, over a 1-orbifold, up to affine equivalence. In this paper, we apply our theory to classify all the geometric fibrations of compact, connected, flat $3$-orbifolds, over a 1-orbifold, up to affine equivalence.
Universidad de la República, Centro de Matemática, Facultad de Ciencias（乌拉圭共和国大学数学中心理学院）；Instituto de Matemática y Estadística, FING, UDELAR（乌拉圭共和国大学工程与信息技术学院数学与统计研究所）；Unidad Académica Matemática, FCEA, UDELAR（乌拉圭共和国大学经济与应用科学学院数学学术单位）
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 totally geodesic, immersed, finite-area surfaces of M called the geometric genus spectrum. They showed that if $M$ is arithmetic and contains a totally geodesic surface, then the geometric genus spectrum of M determines its commensurability class. In this paper we define a coarser inv…
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 braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.
We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios.
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 of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A geometric fibration of E^n/G corre…
