- Series
- Number Theory
- Time
- Wednesday, April 16, 2025 - 3:30pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Tom Tucker – Rochester University – thomas.tucker@rochester.edu – https://people.math.rochester.edu/faculty/ttucker/
- Organizer
- Joshua Stucky
Work of Levin and Przytycki shows that if two non-special rational
functions f and g of degree $> 1 $over $\mathbb{C}$ share the same set of
preperiodic points, there are $m$, $n$, and $r$ such that $f^m g^n = f^r$.
In other words, $f$ and $g$ nearly commute. One might ask if there are
other sorts of relations non-special rational functions $f$ and $g$ over $\mathbb{C}$
might satisfy when they do not share the same set of preperiodic
points. We will present a recent proof of Beaumont that shows that
they may not, that if f and g do not share the same set of preperiodic
points, then they generate a free semi-group under composition. The
proof builds on work of Bell, Huang, Peng, and the speaker, and uses a
ping-pong lemma similar to the one used by Tits in his proof of the
Tits alternative for finitely generated linear groups.