New triangulations show harder skeletons for hyperbolic orbifolds.
problem Embedding tricky skeletons of hyperbolic orbifolds in Euclidean space.
method Generalized Gromov-Guth inequality for hyperbolic n-orbifolds, finding nearly optimal geodesic triangulations.
result Triangulations of skeletons become increasingly difficult to embed nicely in Euclidean space.
Let X be a real analytic orbifold. Then each stratum of X is a subanalytic subset of X. We show that X has a unique subanalytic triangulation compatible with the strata of X. We also show that every Cr-orbifold, 1≤r≤∞, has a real analytic structure. This allows us to triangulate differ…
Geometric methods prove exponential growth in continued fractions.
problem Exponential growth in partial quotients of continued fractions.
method Geometric representation of continued fractions and orbifold triangulations.
result Eventually periodic continued fractions have exponentially growing partial quotients.
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
New homology theories for orbifolds and weighted polyhedra.
problem Homology of orbifolds and weighted polyhedra.
method Introducing AW-homology and DW-homology from special triangulations.
result Invariant under orbifold isomorphisms and generalized Poincaré duality.
The braid group of a complex reflection group is shown to be an index d subgroup.
problem Understanding the structure of braid groups associated with complex reflection groups.
method Presented a compatible presentation for the braid group of the orbifold quotient and a tagged triangulation of the disk.
result The braid group of the complex reflection group G(d,d,n) is an index d subgroup of the braid group of the orbifold quotient. 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…
Paper develops geometry for Kleinian groups using Farey polynomials.
problem Understanding the geometry of Kleinian groups generated by parabolic elements.
method Sakuma-Weeks triangulations and Farey recursive polynomials.
result Simple recursive algorithm to determine link complement geometry.
Associated to any Coxeter system (W,S), there is a labeled simplicial complex L and a contractible CW-complex ΣL (the Davis complex) on which W acts properly and cocompactly. ΣL admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of Sn−1,…
The space of shapes of a polyhedron with given total angles less than 2πat each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a comple…
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
New method shows how certain groups act on 3-orbifolds.
problem Understanding how groups act on 3-dimensional spaces.
method Using veering pairs of laminations and loom spaces.
result Groups with invariant veering pairs are hyperbolic 3-orbifold groups.
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Proves sufficient condition for 2D orbifolds to be good.
problem Characterizing 2D orbifolds as good.
method Analyzes orbifold fundamental groups for goodness.
result Connected 2D orbifolds with infinite orbifold fundamental group are good.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
The paper studies orbifold braid groups and their properties.
problem Understanding orbifold braid groups and their subgroups.
method Detailed study of orbifold braid groups, proving injectivity and triviality of centers.
result Most orbifold braid groups have trivial centers.
Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.
problem Finding shortest non-simple closed geodesics disjoint from orbifold points.
method Fundamental domains and hyperbolic trigonometry.
result Identified and classified all figure eight geodesics on triangle group orbifolds.
Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality and has a natural ring structure. Some examples of orbifold cohomology ring are c…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
The abstract constructs a set of bad 3-orbifolds and shows how any bad 3-orbifold can be transformed into a good one.
problem Characterizing and transforming bad 3-orbifolds into good ones.
method Explicit construction of bad 3-orbifolds and a method of cutting-and-capping to transform them.
result Any bad 3-orbifold can be transformed into a good 3-orbifold through a finite number of operations.
We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .
Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
Minimal triangulations for 229 hyperbolic census knots discovered.
problem Finding minimal triangulations for hyperbolic census knots.
method Ideal triangulations of the magic manifold, low-complexity triangulations for partial fillings, sorting into families.
result Minimal triangulations for 229 hyperbolic census knots discovered, conjectured to be minimal for all 42 families.
Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, givin…
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 …
Study of orbifold Chern character using superconnections.
problem Chern character for orbifold settings.
method Use of flat antiholomorphic superconnections and Riemann-Roch-Grothendieck theorem.
result Uniqueness of orbifold Chern character proven.
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
problem Finding the minimum number of tetrahedra in triangulations of 3-manifolds.
method Computed the triangulation complexity of all elliptic and sol 3-manifolds, within a bounded error.
result Computed the triangulation complexity of all elliptic and sol 3-manifolds.
With the [0,1,2]-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic q-equivelar triangulations of orientable and non-orientable surfaces for every q=3k, k≥2, and every q=3k+1, k≥3. Series of cy…
The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.
problem Understanding the geometric and topological constraints of positively curved Eschenburg orbifolds.
method Proved restrictions on singular sets and computed orbifold cohomology rings.
result Distinctive behavior in cohomology groups of positively curved Eschenburg orbifolds.
We introduce the Γ-Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…
New method connects veering triangulations to dynamic pairs.
problem Understanding veering triangulations and their properties.
method Shearing decomposition of veering triangulations.
result Canonically associated dynamic pairs of branched surfaces.
New loom spaces link flows and triangulations.
problem Understanding flows and triangulations in 3D.
method Introducing loom spaces and proving associated triangulations.
result Locally veering triangulations can be associated to loom spaces.
Highly connected orbifolds are rare but exist.
problem Understanding the maximum connectivity of orbifolds.
method Provided examples and improved bounds for n-connected n-orbifolds.
result Improved bounds for n-connected n-orbifolds, up to dimension 5.
New triangulations encode flows with vanishing polynomial.
problem Constructing veering triangulations with vanishing taut polynomial.
method Using connections between veering triangulations and pseudo-Anosov flows.
result Created arbitrarily large veering triangulations with vanishing taut polynomial.
Orbifold local orientability can be detected by heat invariants.
problem Detecting local orientability in orbifolds.
method Using heat invariants to show Laplace isospectrality.
result Locally orientable orbifolds are distinguishable by heat invariants.