Math 595 – Curve complexes and surface topology

In Fall semester 2014, I taught an advanced topics course on curve complexes at the University of Illinos.

View the course description or course syllabus for an overview.

Office hours:

Tuesdays 10:50-11:50
Wednesdays, 1:30-3:30
and by appointment


Information:

Term: Fall 2014
Time: Tuesdays and Thursdays 9:30-10:50am
Location: 143 Everitt Laboratory

Announcements:

There will be no class Tuesday Dec 9. Instead, our last class will take place on Thursday Dec 11 from 10:30--11:50am in Altgeld 141.

Exercises:

To be presented in class on Oct 9: Suppose that \(\gamma\colon I\to X\) is a path in a metric space \(X\) that there exits a projection \(\pi\colon X\to I\) satisfying the contraction property with constants \(a,b,c>0\) (Definition 2.2 of Masur-Minsky I). Show that there exists a constant \(D\) (depending only on \(a,b,c\)) so that the reverse triangle inequality \[d(\gamma(r),\gamma(t)) \ge d(\gamma(r),\gamma(s)) + d(\gamma(s),\gamma(t)) - D\] holds for all points \(r \le s \le t\) in the interval \(I\subset \mathbb{R}\). (This shows that the path \(\gamma\) is in fact an unparametrized quasi-geodesic: one may precompose by an orientation-preserving homeomorphism of \(I\) so that the result is a (parametrized) \((K,C)\)-quasi-geodesic.)

To be presented in class on Oct 9: On the closed surface of genus 2, find pairs of curves at distance 1, 2 (easy), and 3 (more challenging). If you're feeling ambitious, also find curves at distance 4.

To be presented in class on Oct 9: Show that every quasi-isometry admits a coarse inverse. That is, \(f\colon X\to Y\) is a \((K,C)\) quasi-isometry, show that there exists a \((K',C')\) quasi-isometry \(g\colon Y \to X\) so that \(g\circ f\) and \(f\circ g\) both move points at most distance \(D\), where here \(K', C', D\) only depend on the original constants \(K\) and \(C\).

To be presented in class on Tues Sept 9: Calculate the curve complex of the 4-times punctured sphere, using the redefinition where two essential simple closed curves are adjacent if they intersect twice (which is the smallest possible intersection number in this case). By "calculate" here, I just mean figure out a way to concretely describe it.

Student Presentations:

As desscribed in the course syllabus, I would like to have each student a lecture / presentation on some topic. Here is the ever-evolving list of potential topics: