[Paper Review] Injections of the complex of separating curves into the Torelli complex

Yoshikata Kida|arXiv (Cornell University)|Nov 19, 2009

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper establishes that for all but finitely many compact orientable surfaces with genus $ g \geq 1 $ and Euler characteristic $ |\chi(S)| \geq 4 $, any superinjective map from the complex of separating curves into the Torelli complex is induced by an element of the extended mapping class group. As a consequence, any injective homomorphism from a finite index subgroup of the Johnson kernel into the Torelli group is realized by conjugation via an extended mapping class group element, providing a strong algebraic rigidity result.

ABSTRACT

We show that for all but finitely many compact orientable surfaces, any superinjective map from the complex of separating curves into the Torelli complex is induced by an element of the extended mapping class group. As an application, we prove that any injective homomorphism from a finite index subgroup of the Johnson kernel into the Torelli group for such a surface is induced by an element of the extended mapping class group.

Motivation & Objective

• To establish algebraic rigidity for injective homomorphisms from finite index subgroups of the Johnson kernel into the Torelli group.
• To show that superinjective maps from the complex of separating curves into the Torelli complex are induced by extended mapping class group elements.
• To extend known rigidity results from the mapping class group to the Johnson kernel and Torelli group in the context of curve complexes.
• To resolve the structure of the Torelli group and its subgroups via the geometry of separating curves and bounding pairs.

Proposed method

• Analyzes superinjective maps $ \phi: \mathcal{C}_s(S) \to \mathcal{T}(S) $, where $ \mathcal{C}_s(S) $ is the complex of separating curves and $ \mathcal{T}(S) $ is the Torelli complex.
• Uses the fact that $ \mathcal{C}_s(S) $ is a subcomplex of $ \mathcal{T}(S) $ and that the extended mapping class group $ \mathrm{Mod}^*(S) $ acts naturally on both complexes.
• Applies results from [15] on superinjective maps on $ \mathcal{C}_s(S) $ to show that such maps are induced by group elements.
• Employs a chain of normal subgroups $ \mathcal{K}(S) = N_0 < \cdots < N_{p-1} = \mathcal{I}(S) $ with abelian quotients to analyze the structure of the Torelli group.
• Uses the relative commensurator and abstract commensurator to analyze injective homomorphisms between finite index subgroups.
• Applies the conjugation action of $ \mathrm{Mod}^*(S) $ to show that any such injective homomorphism is realized by conjugation.

Experimental results

Research questions

• RQ1 Under what conditions is a superinjective map from the complex of separating curves into the Torelli complex induced by an element of the extended mapping class group?
• RQ2 Can every injective homomorphism from a finite index subgroup of the Johnson kernel into the Torelli group be realized as conjugation by an element of the extended mapping class group?
• RQ3 What structural properties of the Torelli group and Johnson kernel allow such rigidity results to hold?
• RQ4 How do the normal subgroup chains in the Torelli group influence the behavior of automorphisms and injective homomorphisms?
• RQ5 What is the role of the Euler characteristic and genus in determining the validity of such rigidity theorems?

Key findings

• For all but finitely many compact orientable surfaces with $ g \geq 1 $ and $ |\chi(S)| \geq 4 $, any superinjective map $ \phi: \mathcal{C}_s(S) \to \mathcal{T}(S) $ satisfies $ \phi(\mathcal{C}_s(S)) \subset \mathcal{C}_s(S) $.
• Any superinjective map $ \phi: \mathcal{C}_s(S) \to \mathcal{T}(S) $ is induced by an element $ \gamma \in \mathrm{Mod}^*(S) $, so $ \phi(a) = \gamma a $ for all vertices $ a \in \mathcal{C}_s(S) $.
• Any injective homomorphism $ f: \Gamma \to \mathcal{I}(S) $, where $ \Gamma $ is a finite index subgroup of the Johnson kernel $ \mathcal{K}(S) $, is realized as conjugation by some $ \gamma_0 \in \mathrm{Mod}^*(S) $.
• The relative commensurator of $ \mathcal{K}(S) $ in $ \mathrm{Mod}^*(S) $ acts via conjugation on finite index subgroups, and such homomorphisms are fully determined by this action.
• For surfaces with $ g \geq 3, p \leq 1 $ or $ g = 1, p \geq 4 $, any injective endomorphism $ f: \Gamma \to \Gamma $ with $ \Gamma $ of finite index in $ \mathrm{Mod}^*(S) $ and satisfying $ [\mathcal{K}(S) : \Gamma \cap \mathcal{K}(S)] < \infty $, $ [\Gamma : \Gamma \cap \mathcal{I}(S)] < \infty $, satisfies $ \gamma \Gamma \gamma^{-1} = \Gamma $ for some $ \gamma \in \mathrm{Mod}^*(S) $.
• The Torelli group $ \mathcal{I}(S) $ admits a finite normal series with abelian quotients, and this structure is preserved under conjugation by elements of $ \mathrm{Mod}^*(S) $, enabling the proof of rigidity.