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| pants complex | |
|---|---|
| Name | Pants complex |
| Type | Simplicial complex |
pants complex
The pants complex is a combinatorial and geometric object associated to a compact surface that encodes decompositions of the surface into pairs of pants. It serves as a tool connecting Teichmüller space, mapping class group, and low-dimensional topology, and it appears in the study of hyperbolic geometry and 3-manifold theory. The complex supports actions by mapping class groups of surfaces such as those studied by William Thurston, Benson Farb, and Howard Masur.
The pants complex is defined for a compact, oriented surface S_{g,n} of genus g with n punctures and its vertices correspond to isotopy classes of pants decompositions represented by collections of disjoint simple closed curves on S_{g,n} that cut it into three-holed spheres. Vertices are connected by edges encoding elementary moves such as the elementary pants move (sometimes called an A-move or B-move) related to the Farey graph and Dehn twist operations by elements of the mapping class group. The complex is locally finite for finite-type surfaces, connected, and often equipped with a graph metric making it a path metric space on which the mapping class group of S_{g,n} acts by simplicial automorphisms.
Construct a pants complex by taking the 0-skeleton as isotopy classes of pants decompositions; add 1-simplices when two decompositions differ by a single elementary move supported on a four-holed sphere or a one-holed torus. Higher dimensional simplices record commuting sequences of moves, similar to cube complexes appearing in the work of Masur–Minsky and the cubical structures studied by D. Margalit and Yair Minsky. Combinatorially, the local link of a vertex decomposes into subgraphs corresponding to curve complexes of subsurfaces like the curve complex of an annulus and the arc complex for punctured disks; these combinatorial patterns mirror complexes studied by John Hempel and Herbert Masur.
Equipping the 1-skeleton with the graph metric yields a metric space quasi-isometric to other complexes of interest such as the Weil–Petersson metric completion of Teichmüller space in many contexts studied by Brock, Wolpert, and Jeffrey Brock. The pants complex exhibits large-scale hyperbolicity phenomena analogous to those in the curve complex as in results by Masur and Minsky. Distances in the pants complex coarsely correspond to length-spectrum and twisting distances studied in Teichmüller theory and the geometry of hyperbolic surfaces investigated by Colin R. de Verdière and Maryam Mirzakhani.
The mapping class group of S_{g,n} acts properly discontinuously by isometries on the pants complex, yielding a combinatorial model for studying group properties such as quasi-isometric rigidity and Nielsen–Thurston classification of mapping classes like pseudo-Anosov, reducible, and periodic types explored by Nielsen and Thurston. The pants complex provides coarse coordinates for Teichmüller space analogous to Fenchel–Nielsen coordinates and interfaces with the Weil–Petersson metric through quasi-isometries described in work of Brock and Masur–Minsky. Relations to earthquake maps studied by Kerckhoff and to length-spectrum metrics considered by Lenzhen appear through projection maps from pants decompositions to measured laminations investigated by Bonahon and Papadopoulos.
For low-complexity surfaces like the four-punctured sphere S_{0,4} and the one-holed torus S_{1,1}, the pants complex reduces to classical graphs such as the Farey graph and is explicitly computable using continued fraction combinatorics linked to Series and Bowditch. For S_{0,4}, vertices correspond to three-punctured sphere decompositions with adjacency modeled on elementary moves studied in Floyd and Parry. For genus two surfaces S_{2,0}, computations involve subsurface projections to curve complexes of one-holed tori and four-holed spheres and use techniques from Masur–Minsky hierarchical structure and work of Aougab.
The pants complex is used to compare combinatorial and analytic structures: it appears in proofs of quasi-isometric models for Weil–Petersson metric by Brock, constructions of Heegaard splittings in 3-manifold topology examined by Moriah and Scharlemann, and in the study of hyperbolic 3-manifolds via ending lamination conjectures by Brock–Canary–Minsky and Thurston. It also informs algorithmic problems such as computing distances in complexes for mapping class group algorithms studied by Holt and Bell, and intersects with the study of surface bundles examined by Farb–Margalit.
Variants include the pants graph, augmented pants complexes with marking data as in Masur–Minsky hierarchies, and complexes decorated by lengths or twist parameters connected to Fenchel–Nielsen coordinates and the augmented Teichmüller space of Abikoff and Bers. Higher-dimensional generalizations consider complexes of decompositions into other pieces studied in relation to the Hatcher–Thurston complex and work on sphere systems by Hatcher and McCullough. Recent expansions incorporate coarse median structures and connections to cube complexes investigated by Bowditch and Behrstock.