Mapping class groups (Math 851), Fall 2011

Tuesday and Thursday 11AM - 12:15 in Van Vleck B309

with Autumn Kent.

Office: Van Vleck 615

Office hours: Tuesday 2-4, but I have an open door policy. Come on by!

Text: A primer on mapping class groups, by Farb and Margalit.

Resources:

For an introduction to conformal mappings, see the book

An introduction to complex function theoryby Bruce Palka

For introductions to hyperbolic geometry and hyperbolic surfaces, see the following texts:

Fuchsian groupsby Svetlana Katok

and

Teichmüller Theoryby John Hubbard

Problems:

I will occasionally post some problems here. While I will not collect solutions, we may discuss the problems in class. Be warned that I may intentionally pose problems that are open or false. Note also that I list problems in reverse chronological order.

16.

15. Prove that if \(\alpha\) and \(\beta\) are two disjoint homotopic simple closed curves in a surface \(S\), then there is an annulus \(A \cong \mathbb{S}^1 \times I\) in \(S\) whose boundary is \(\alpha \cup \beta\).

14. Let \(\gamma\) and \(\delta\) be hyperbolic elements in \(\mathrm{PSL}_2(\mathbb{R})\) whose fixed point sets in \(\mathbb{S}^1_\infty\) are disjoint. Show that there is a natural number \(n\) such that \[ \left\langle \gamma^n, \delta^n \right\rangle \] is isomorphic to a free group.

13. Show that a metric \[ \mathrm{d}s^2 = E\, \mathrm{d}u^2 + 2F\, \mathrm{d}u \, \mathrm{d}v + G \, \mathrm{d}v^2 \] on a domain in \(\mathbb{C}\) (where \(z = u + \mathrm{i}v\)) is conformal if and only if the functions \(E\), \(F\), and \(G\) satisfy \(E = G\) and \(F=0\).

12. Show that \[ \left\langle \frac{z + 1}{z+2}, \frac{1-z}{z-2} \right\rangle \] is a discrete subgroup of \(\mathrm{PSL}_2(\mathbb{R}) \).

11. Show that there are only four essential simple closed curves on the Klein bottles, up to isotopy.

10. (Geodesic in every direction) Let \(S\) be a closed hyperbolic surface. A (local) geodesic ray \(r : \mathbb{R} \to S\) may always be lifted to a path in the unit tangent bundle \(\mathrm{T}^1(S)\). Show that there is a geodesic ray \(r : \mathbb{R} \to S\) whose lift is dense in \(\mathrm{T}^1(S)\).

9. Show that an orientation preserving isometry of \(\mathbb{H}^2\) is elliptic, parabolic, or hyperbolic precisely when its trace has absolute value less than two, equal to two, or greater than two, respectively.

8. Let \(a\), \(b\), and \(c\) be natural numbers. Show that there is a finite group \(G\) containing elements \(x\) and \(y\) such that the orders of \(x\), \(y\), and \(xy\) are \(a\), \(b\), and \(c\), respectively.

7. An orientation reversing isometry of \(\mathbb{H}^2\) is called areflectionif it fixes a geodesic pointwise. Show that any isometry of \(\mathbb{H}^2\) is a product of one, two, or three reflections.

6. Show that circles in \(\mathbb{H}^2\) are the same as Euclidean circles, but with different centers.

5. Let \(x\) and \(y\) be points in \(\mathbb{H}^2\). Show that the geodesic joining them is unique.

4. Your Heegaard splittings \(H_1 \cup H_2\) from #3 have the property that there are disks \( (D_1,\partial D_1) \) and \( (D_2,\partial D_2) \) embedded (as pairs) in \( (H_1, \partial H_1 ) \) and \( (H_2, \partial H_2 ) \), respectively, so that \(\partial D_1 \cap \partial D_2\) is a single point. This is called areducibleHeegaard splitting. Can you find a \(3\)-manifold with infinitely many irreducible Heegaard splittings?

3. Show that every closed orientable \(3\)-manifold \(M\) has infinitely many Heegard splittings, up to homeomorphism of \(M\).

2. A surface isplanarif it may be embedded in the \(2\)-sphere. Show that a planar surface \(P\) is either the \(2\)-sphere, the disk, an annulus, or has a finite cover that is not planar.

1. (Hauptvermutung for surfaces)

Atriangulation\(T\) of a space \(X\) is a simplicial complex \(T\) and a homeomorphism \(T \to X\). Asubdivision\(D\) of a triangulation \(T \to X\) is a triangulation \(D \to X \) such that the image of each simplex of \(T\) is a union of images of simplices of \(D\). Two simplicial complexes areisomorphicif they are homeomorphic via a map that takes simplices to simplices via linear homeomorphisms. Two triangulations \(f: T_1 \to X \) and \(f: T_2 \to X \) of a space \(X\) are said to beequivalentif \(T_1\) and \(T_2\) have isomorphic subdivisions.

Let \(S\) be a closed surface. Show that any two triangulations of \(S\) are equivalent. (As a warm up, show that any two triangulations of the disk are equivalent.)

0. (Klyachko's car crash lemma) Suppose that the 2-sphere is equipped with a cell structure such that each closed 2-cell is embedded. On the boundary of each 2-cell, there is a taxi driving counterclockwise (from the perspective of the 2-cell), and it will travel around the boundary infinitely many times. Show that there is a crash.