[Paper Review] Point pushing, homology, and the hyperelliptic involution

Tara Brendle, Dan Margalit|arXiv (Cornell University)|Oct 6, 2011

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper establishes a Birman exact sequence for the hyperelliptic Torelli group, proving it splits in two key cases: when marking a fixed point or a pair of swapped points under the hyperelliptic involution. The splitting allows the authors to show the hyperelliptic Torelli group is generated by Dehn twists about symmetric separating curves if and only if it is generated by reducible elements, resolving a conjecture in part and linking to the kernel of the Burau representation.

ABSTRACT

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As a consequence, we show that the hyperelliptic Torelli group is generated by Dehn twists if and only if it is generated by reducible elements. We also give an application to the kernel of the Burau representation.

Motivation & Objective

• To extend the Birman exact sequence to the hyperelliptic Torelli group, a subgroup of the mapping class group acting trivially on homology and commuting with a hyperelliptic involution.
• To understand the structure of the hyperelliptic Torelli group via exact sequences involving marked surfaces and boundary components.
• To resolve the relationship between generation by Dehn twists and generation by reducible elements in the hyperelliptic Torelli group.
• To apply the results to the kernel of the Burau representation, particularly in the context of symmetric curve structures.

Proposed method

• Construct a Birman exact sequence for the hyperelliptic Torelli group by considering surfaces with marked points or boundary components.
• Identify the kernel of the forgetful map from the marked hyperelliptic Torelli group to the unmarked group using fundamental group data from a sphere with punctures.
• Use the Birman–Hilden theorem to realize the kernel as a subgroup of the free group on 2g+1 generators, with a homomorphism ε to ℤ^{2g+1} defined on even-length words.
• Prove the exact sequence splits by constructing a section using symmetric and pre-symmetric curves, leveraging the fact that the kernel of ε is an infinitely generated free group.
• Apply the splitting to show that elements in the kernel are products of Dehn twists about symmetric separating curves.
• Use induction on genus, combining the Birman sequence with the structure of symmetrically reducible elements to prove generation by symmetric Dehn twists.

Experimental results

Research questions

• RQ1 Does the hyperelliptic Torelli group admit a Birman exact sequence analogous to the standard mapping class group?
• RQ2 Is the kernel of the forgetful map from the marked hyperelliptic Torelli group to the unmarked group isomorphic to a free group, and does the sequence split?
• RQ3 Can the hyperelliptic Torelli group be generated by Dehn twists about symmetric separating curves if and only if it is generated by reducible elements?
• RQ4 What is the role of symmetric and pre-symmetric curves in constructing sections of the Birman sequence?
• RQ5 How does the structure of the kernel relate to the kernel of the Burau representation?

Key findings

• The forgetful map from the hyperelliptic Torelli group with a single fixed marked point to the unmarked group is an isomorphism, so the Birman sequence degenerates.
• For a surface with one boundary component, the hyperelliptic Torelli group is isomorphic to the product of the unmarked group and ℤ.
• When marking a pair of points swapped by the hyperelliptic involution, the kernel of the forgetful map is isomorphic to the kernel of a homomorphism ε from the even-length words in a free group of rank 2g+1 to ℤ^{2g+1}.
• The resulting short exact sequence is split, so the marked hyperelliptic Torelli group is isomorphic to the semidirect product of the unmarked group with an infinitely generated free group.
• Every element in the kernel of the forgetful map is a product of Dehn twists about symmetric separating curves.
• The hyperelliptic Torelli group is generated by Dehn twists about symmetric separating curves if and only if it is generated by reducible elements, as shown via induction on genus and the use of the Birman sequence.