Math 9200 Mapping Class Groups Fall 2025 -	-----------------------------------Sullivan & Thurston

Here are the notes from this semester.

Office Hours

• 1.1 Give one interesting example, and one interesting non-example, of the change of coordinates principle.
• 1.2 How many simply closed curves are there, up to homeomorphism, in a closed, non-orientable surface?
• 2.1 Show that that Mod^±(S_0,4) is the semidirect product of PGL₂(Z)[2] with S₄
• 2.2 Use the Alexander Method to verify the daisy relation in Mod(S_0,5).
• 3.1 Write down a matrix for the action on homology of an element of order 4g+2 in Mod(S_g). What is the order of the matrix? What is the characteristic polynomial?
• 3.2 Draw an oriented curve in S₃ representing the homology class (1,3,-2,2,-1,1). Use the standard basis.
• E.1 Find two generators for the automorphism group of a surface group.
• 4.1 Using the second and third definitions of the Johnson homomorphism, directly compute the images of a Dehn twist about separating curve and a bounding pair map.

Final project ideas

• Which sequences a_n can (or cannot) be obtained as $\left(i\left(f^n\right(c),d)\right)$ for some surface S, some curve/arc c and some curve/arc d?
• Understand this paper and extend the ideas: https://arxiv.org/pdf/2508.16323
• Use the ping pong lemma to make more free groups. For instance, do non-commuting bounding pair maps always generate a free group of rank 2? Do any two elements of Torelli group either commute or generate a free group of rank 2?
• Is the mapping class group generated by an element of order 2 and an element of order 3?
• What possible numbers of fixed points are there for a periodic element of Mod(S_g)?
• Is there another generating set for Mod(S_g) that has 2g+1 Dehn twists and is not equivalent to the Humphries generating set?
• How many conjugates of a given pseudo-Anosov mapping class are needed to generate Mod(S_g)?
• Can we describe all of the periodic elements in Mod(S₅)?
• Describe the Torelli group corresponding to the action of Mod(S_g^b) on cohomology.
• Write down explicit finite generating sets for Aut pi₁(S_g).
• Find an explicit free generating set for I(S₂).

Resources

• Casson & Bleiler (includes introduction to the hyperbolic plane)