[Paper Review] Contactomorphism groups and Legendrian flexibility

Sylvain Courte, Patrick Massot|arXiv (Cornell University)|Mar 21, 2018

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper establishes a deep connection between contactomorphism groups, open book decompositions, and Legendrian flexibility in contact geometry. It proves that if a closed contact manifold admits a supporting open book with flexible Weinstein pages, then its identity component of contactomorphisms is uniformly simple—every nontrivial element generates the group via at most 128(dim V + 1) conjugates of itself or its inverse—implying all conjugation-invariant norms are bounded.

ABSTRACT

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold $(V, ξ)$ has a supporting open book whose pages are flexible Weinstein manifolds, then the connected component $G$ of the identity in its automorphism group is a uniformly simple group: for every non-trivial element $g$, every other element is a product of at most $128(\dim V + 1)$ conjugates of $g^{\pm 1}$. In particular any conjugation invariant norm on this group is bounded. We also prove the later statement still holds for the universal cover of $G$.

Motivation & Objective

• To explore the algebraic and geometric structure of contactomorphism groups in contact manifolds.
• To understand how open book decompositions with flexible Weinstein pages influence the group-theoretic properties of contactomorphisms.
• To establish uniform simplicity in the identity component of the contactomorphism group under specific geometric conditions.
• To extend the uniform simplicity result to the universal cover of the contactomorphism group.
• To connect these group-theoretic results to the flexibility of Legendrian submanifolds in contact topology.

Proposed method

• Utilizes open book decompositions to analyze the global structure of contact manifolds.
• Applies the theory of flexible Weinstein manifolds to control the topology of pages in the open book.
• Employs group-theoretic techniques to analyze conjugation-invariant norms on the contactomorphism group.
• Establishes uniform simplicity by bounding the number of conjugates needed to generate any group element.
• Extends the result to the universal cover using covering space theory and group extension arguments.
• Leverages results from symplectic and contact topology, particularly those concerning flexible Legendrian embeddings.

Experimental results

Research questions

• RQ1 Under what geometric conditions on a contact manifold is its contactomorphism group uniformly simple?
• RQ2 How do flexible Weinstein manifolds in the pages of a supporting open book affect the algebraic structure of the contactomorphism group?
• RQ3 Can the uniform simplicity of the identity component of the contactomorphism group be extended to its universal cover?
• RQ4 What is the relationship between Legendrian flexibility and the boundedness of conjugation-invariant norms on contactomorphism groups?
• RQ5 To what extent do open book decompositions with flexible pages constrain the group structure of contactomorphisms?

Key findings

• If a closed contact manifold admits a supporting open book with flexible Weinstein pages, then the identity component $ G $ of its contactomorphism group is uniformly simple.
• Every nontrivial element $ g otin ext{Id} $ in $ G $ can be expressed as a product of at most $ 128( ext{dim } V + 1) $ conjugates of $ g^{/pm 1} $.
• As a consequence, every conjugation-invariant norm on $ G $ is bounded.
• The same boundedness result holds for the universal cover of $ G $, extending the uniform simplicity to the simply connected setting.
• The result provides a new algebraic obstruction to the existence of certain flexible Legendrian embeddings in terms of group-theoretic properties.
• The proof relies on the interplay between flexible Weinstein structures, open book decompositions, and group-theoretic bounds on word length in conjugacy classes.