The pants graph of a free group is constructed and studied.
problem Understanding the structure of free groups through graph theory.
method Developed a pants graph and studied its properties.
result The pants graph of a free group is connected and unbounded.
Construct special Lagrangian pair of pants in n dimensions.
problem Constructing special Lagrangian pair of pants in general dimensions.
method Inside the cotangent bundle of Tn with the Euclidean structure. result Special Lagrangian pair of pants constructed in general dimensions.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in CPn+1 decomposes into pairs of pants: a pair of pants is a real compact 2n-manifold with cornered boundary obtained by removing an open regular neighborhood of n+2 generic hyperplanes from CPn. As is we…
Long systoles form pants decompositions on hyperbolic surfaces.
problem Constructing hyperbolic surfaces with long systoles.
method Construction of sequences of closed hyperbolic surfaces.
result Systoles form pants decompositions and their length grows logarithmically with genus.
Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the 1-skeleton of the pants complex.
The pants graph of a non-orientable surface is quasi-isometric to its Teichmüller space.
problem Understanding the relationship between pants graphs and Teichmüller spaces of non-orientable surfaces.
method Constructing a map between pants graphs induced by lifting pants decompositions and proving quasi-isometric embeddings.
result The pants graph of a non-orientable surface is quasi-isometric to its Teichmüller space.
We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any n≥4, we construct a finite subgraph Xn of the pants graph P(S0,n) of the n-punctured sphere S0,n with the following property. Any simplicial embedding of Xn into any pants graph P(S0,m) of a punctured …
Researchers exhaustively cover pants graphs of spheres with punctures using finite rigid sets.
problem Covering all pants graphs of punctured spheres efficiently.
method Constructing a sequence of finite rigid sets in the pants graph.
result The union of these sets covers the entire pants graph of the punctured sphere.
In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.
Developing tools for computing string amplitudes with hyperbolic vertices.
problem Computing off-shell string amplitudes with new vertices.
method Constructing local coordinates and investigating limits for hyperbolic three-string vertex.
result Derived conservation laws and performed sample computations.
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, in a natural way its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…
3-manifolds virtually 1-dominated by hyperbolic ones.
problem Domination of 3-manifolds by hyperbolic ones.
method Enhanced connection principle in good pants constructions.
result Every 3-manifold is virtually 1-dominated by a hyperbolic one.
Let Σ be a surface of negative Euler characteristic together with a pants decomposition ¶. Kra's plumbing construction endows Σ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctu…
The abstract discusses coordinates for convex projective structures on surfaces.
problem Describing coordinates for convex projective structures on surfaces.
method Goldman and Bonahon-Dreyer constructed two sets of global coordinates for P(S). The article explicitly describes the coordinate change between these two parametrizations.
result Explicit description of coordinate change between Goldman and Bonahon-Dreyer coordinates.
This paper calculates hyperbolicity constants for pants and relative pants graphs.
problem Understanding how hyperbolicity constants change based on the surface.
method Study of hyperbolicity constants for pants and relative pants graphs for specific surfaces.
result Calculates hyperbolicity constants for the five-punctured sphere and twice punctured torus.
We consider a planar surface Σof infinite type which has the Thompson group T as asymptotic mapping class group. We construct the asymptotic pants complex C of Σand prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the …
Let Sg denote the genus g closed orientable surface. For k∈N, a k-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than k times. Juvan-Malnič-Mohar \cite{Ju-Mal-Mo} showed that there exists a k-system on Sg whose size is on the order o…
New invariants from trisections compare 4-manifolds.
problem Comparing 4-manifolds using trisections.
method Adapting Heegaard splitting techniques to trisections, constructing invariants from distances in pants complexes.
result Invariants are independent of choices and interpret nearby manifolds.
We consider a union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to some handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3-manifold. We introduce a groupoid FT acting on double pants …
Connected graph for twice-punctured torus curves.
problem Structure of tri-pants graph on twice-punctured torus.
method Examined relationship with Farey complex to prove connectivity and infinite diameter.
result Tri-pants graph is connected and has infinite diameter.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
The purpose of this paper is to establish an upper bound on the distance between two pants decompositions in the pants complex for a closed surface of genus g >= 2. This is done by use of graph theory. First distance is found in the pants graph modulo the action of the mapping class group, and then between pants decomp…
Study of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds.
problem Understanding the structure of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds. method Investigating the abelian group generated by (R,ε)-good curves in M modulo the oriented boundaries of (R,ε)-good pants. result For sufficiently small ε>0 and sufficiently large R>0, the (R,ε)-panted cobordism group of M is isomorphic to H1(extSO(M);Z). Formula calculates surface torsion via pants decompositions.
problem Computing Reidemeister torsion of compact surfaces.
method Used a specific pants decomposition to compute torsion.
result Formula for surface torsion in terms of pairs of pants.
Two Morse theory algorithms segment surface meshes into pants.
problem Segmenting surface meshes into pants.
method Two Morse theory algorithms: one using handles, the other using Reeb graph.
result Reeb graph based algorithm runs faster and achieves best time efficiency.
Study examines equivalence relations on the pair of pants, proving k-equivalence implies 1-equivalence and 2-equivalence.
problem Understanding equivalence classes of closed curves on the pair of pants.
method Examined k-equivalence, proving it implies 1-equivalence and 2-equivalence, and deepened understanding of 1-equivalence.
result k-equivalence implies 1-equivalence and 2-equivalence on the pair of pants.
Smooth 4-manifolds can be represented by loops in the pants complex.
problem Representing smooth 4-manifolds using a specific geometric structure.
method Using loops in the pants complex to represent and analyze 4-manifolds.
result Smooth 4-manifolds are smoothly cobordant to ∐mCP2∐nCPˉ2. Study automorphisms on procongruence curve and pants complexes.
problem Understanding automorphism groups of procongruence curve and pants complexes.
method Action on procongruence mapping class group and rigidity theorem for pants complex.
result Prove rigidity theorem for procongruence completion of pants complex.
Study shows pants graph automorphisms match mapping class groups of nonorientable surfaces.
problem Understanding automorphisms of pants graphs on nonorientable surfaces.
method Analyzing mapping class groups and proving isomorphism.
result Automorphism group of pants graphs isomorphic to mapping class groups.
The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.
problem Computing the number of self-intersections of closed geodesics on a pair of pants.
method Used Bowen and Series' coding to compute self-intersections.
result Proved a conjecture and provided bounds for self-intersection numbers.
Let X be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space T(X) of the surface X using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…
We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class grou…
We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces (the good pants are pairs of pants whose cuffs have the length nearly equal to some large number R). Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpre…
We show that for a surface S, the subgraph of the pants graph determined by fixing a collection of curves that cut S into pairs of pants, once-punctured tori, and four-times-punctured spheres is totally geodesic. The main theorem resolves a special case of a conjecture made by Aramayona, Parlier, and Shackleton and has…
Researchers find a finite rigid subgraph in pants graphs of surfaces.
problem Identifying a finite rigid subgraph in pants graphs of surfaces.
method Identifying a finite rigid subgraph Xg,n of pants graph P(Sg,n). result Any simplicial embedding of Xg,n into any pants graph P(Sg′,n′) is induced by an embedding Sg,noSg′,n′. The paper finds a special pants decomposition for certain surface group representations.
problem Finding a specific pants decomposition for surface group representations.
method Proves the existence of a pants decomposition with irreducible restrictions and no trace ±2 elements.
result Shows the existence of a special pants decomposition for SL2(C)-representations of surface groups. This paper classifies fibered links in 3-sphere using open book decompositions.
problem Classifying fibered links in the 3-sphere.
method Constructing links and their pair-of-pants fiber surfaces from a Hopf link, then verifying monodromies using Heegaard diagrams.
result Monodromies of the links in the family are the only ones corresponding to pair-of-pants open book decompositions of the 3-sphere.
4-manifolds can be broken down into pairs-of-pants and K3 surfaces.
problem Decomposing 4-manifolds diffeomorphic to complex hypersurfaces.
method Pair-of-pants and K3 surfaces decomposition.
result 4-manifolds diffeomorphic to complex hypersurfaces can be decomposed into a specific number of pair-of-pants and K3 surfaces.
Characterizes lamination spaces of graphs on a pair of pants.
problem Understanding lamination spaces of graphs on a pair of pants.
method Identifying lamination spaces as lattice polytopes and using graph exploration technique.
result Characterizes the polytopes that arise as lamination spaces of graphs on a pair of pants.
Study Fuchsian loci in mPSLn(R)-Hitchin components of a pair of pants.
problem Understanding Fuchsian loci in mPSLn(R)-Hitchin components. method Using Bonahon-Dreyer parametrization, explicit parametrization of Fuchsian loci of a pair of pants.
result Explicit parametrization of Fuchsian loci of a pair of pants.
The paper bounds distances and transformations between pants decompositions and triangulations on surfaces.
problem Finding bounds on distances and transformations between pants decompositions and triangulations.
method Using pre-triangulations, train tracks, and Agol-Hass-Thurston algorithm.
result Upper bounds on distances and transformations between pants decompositions and triangulations.
Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
P-moves connect different 3-manifold decompositions.
problem Connecting different 3-manifold decompositions.
method Using P-complex and Morse 2-functions, we show P-moves between pants-block decompositions.
result Any two pants-block decompositions are related by a finite sequence of P-moves.
A family of interpolating graphs $\calC (S, ξ)$ of complexity ξ is constructed for a surface S and −2≤ξ≤ξ(S). For ξ=−2,−1,ξ(S)−1 these specialise to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalise Theorems of Brock-Farb and Behrstock-Mins…
Improves bounds on surface decompositions.
problem Surface decomposition bounds
method Improved theorem of Bers for surfaces with boundary and closed surfaces.
result Slight improvement on best known bound for closed surfaces.
We prove a number of convexity results for strata of the diagonal pants graph of a surface, in analogy with the extrinsic geometric properties of strata in the Weil-Petersson completion. As a consequence, we exhibit convex flat subgraphs of every possible rank inside the diagonal pants graph.
Study pants' terms on hyperbolic surfaces, proving length dominance implies isometry and pants bounds.
problem Understanding terms in hyperbolic surfaces' identities and their implications.
method Analyzing properties of terms and applying convexity to deduce theorems.
result Proved that if a simple length spectrum dominates another, the surfaces are isometric, and found bounds on pants numbers.