[Paper Review] The automorphism group of the pants complex

Dan Margalit|ArXiv.org|Jan 31, 2002

Geometric and Algebraic Topology8 references11 citations

TL;DR

This paper establishes that the automorphism group of the pants graph—the 1-skeleton of the complex of pants decompositions—is isomorphic to the extended mapping class group of a closed, oriented surface of negative Euler characteristic, excluding the torus with two punctures. The proof relies on characterizing 2-cells combinatorially within the 1-skeleton and linking marked Farey graphs in the pants graph to vertices in the curve complex, thereby inducing an isomorphism between automorphism groups.

ABSTRACT

We show that the automorphism group of the complex of pants decompositions for a surface is isomorphic to the mapping class group for that surface.

Motivation & Objective

• To establish a combinatorial model for the mapping class group using the pants complex.
• To show that the automorphism group of the pants graph (the 1-skeleton of the pants complex) is isomorphic to the extended mapping class group.
• To demonstrate that the topological structure of the pants complex can be fully recovered from its 1-skeleton via combinatorial relations.
• To link automorphisms of the pants graph to automorphisms of the curve complex through marked Farey graphs.
• To prove that all automorphisms of the pants graph arise from mapping class group elements.

Proposed method

• Characterize 2-cells in the pants complex combinatorially using loops of four edges in the pants graph that do not lie in a common Farey graph.
• Use the fact that square 2-cells correspond to commutator relations of elementary moves on disjoint subsurfaces, and show this property is purely combinatorial.
• Define marked Farey graphs in the pants graph as collections of pants decompositions differing by a single curve, with a common reference point.
• Establish a correspondence between vertices in the curve complex and marked Farey graphs in the pants graph.
• Construct a map from automorphisms of the pants graph to automorphisms of the curve complex by preserving intersection patterns of Farey graphs.
• Prove that the induced map is an isomorphism by showing it is a well-defined homomorphism, injective, and surjective.

Experimental results

Research questions

• RQ1 Can the automorphism group of the pants complex be fully described using only the 1-skeleton of the complex?
• RQ2 Is every automorphism of the pants graph induced by an element of the extended mapping class group?
• RQ3 Can the 2-cell structure of the pants complex be reconstructed from the combinatorics of the pants graph alone?
• RQ4 How do marked Farey graphs in the pants graph relate to vertices in the curve complex?
• RQ5 Is the automorphism group of the pants graph isomorphic to the mapping class group of the surface?

Key findings

• The automorphism group of the pants graph is isomorphic to the extended mapping class group of the surface, provided the surface is not the genus 1 surface with two punctures.
• The 2-cell structure of the pants complex can be characterized entirely within the 1-skeleton using combinatorial conditions on 4-cycles that do not lie in a common Farey graph.
• Automorphisms of the pants graph preserve the structure of marked Farey graphs, which correspond bijectively to vertices in the curve complex.
• The induced map from the automorphism group of the pants graph to the automorphism group of the curve complex is an isomorphism.
• The isomorphism between the automorphism group of the pants graph and the mapping class group is natural and arises from the action of mapping classes on pants decompositions.
• The result confirms that the pants graph is a canonical combinatorial model for the Teichmüller space with the Weil-Petersson metric, as its automorphism group matches the isometry group of the space.