NICK SALTER

Geometry, Topology, and the Torelli group

The goal of this course is to explore mathematics in and around the Torelli group, with a particular emphasis on finiteness and non-finiteness properties. For specialists, the course will cover Johnson's contributions to the subject, as well as later developments, culminating in the recent proof of Ershov-He that the Johnson kernel is finitely generated. For non-specialists, the subject material will serve as a vehicle to visit a lot of beautiful and insightful mathematics, and will touch on 3-manifold topology, geometric group theory, and algebraic geometry.

• Introduction to the Torelli group and the Johnson filtration.
  • Farb-Margalit, chapter 6
  • Johnson, An abelian quotient of I_g
  • Johnson, A survey of the Torelli group
  
  
• Applications of the Johnson homomorphism.
  • Broaddus-Farb-Putman, The Casson invariant and the word metric on the Torelli group
  • Broaddus-Farb-Putman, Irreducible Sp-representations and subgroup distortion in the mapping class group
  
  
• Finite generation of the Torelli group.
  • Johnson, The structure of the Torelli group I
  
  
• Z/2 quotients of the Torelli group.
  • Johnson, Quadratic forms and the Birman-Craggs homomorphisms
  • Brendle-Farb, The Birman-Craggs-Johnson homomorphism and abelian cycles in the Torelli group
  
  
• The Johnson kernel and separating twists.
  • Johnson, The structure of the Torelli group II
  • Putman, The Johnson homomorphism and its kernel
  
  
• The abelianization of the Torelli group.
  • Johnson, The structure of the Torelli group III
  
  
• Some non-finiteness results
  • Akita, Homological infiniteness of Torelli groups
  • Church-Farb, Infinite generation of the kernels of the Magnus and Burau representations
  • Bestvina-Bux-Margalit, The dimension of the Torelli group
  
  
• The Torelli group in genus 2
  • Mess, The Torelli groups for genus 2 and genus 3 surfaces
  • McCullough-Miller, The genus 2 Torelli group is not finitely generated
  • Hatcher-Margalit, Generating the Torelli group
  
  
• Representation stability and the Torelli group
  • Church-Farb, Parametrized Abel-Jacobi maps and abelian cycles in the Torelli group
  • Church-Putman, Generating the Johnson filtration
  • Boldsen-Dollerup, Towards representation stability for the second homology of the Torelli group
  
  
• BNS invariants
  
• Resonance varieties and cohomology jump loci
  
• The abelianization of the Johnson kernel
  • Dimca-Papadima, Arithmetic group symmetry and finiteness properties of Torelli groups
  • Dimca-Hain-Papadima, The abelianization of the Johnson kernel
  
  
• Finite generation of the Johnson kernel.
  • Ershov-He, On finiteness properties of the Johnson filtrations

Other topics/ supplementary references

• Hain, Infinitesimal presentations of the Torelli group
• Deligne-Griffiths-Morgan-Sullivan, Real homotopy theory of Kahler manifolds