Relations between rational functions and an analog of the Tits alternative

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.eduhttps://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.