Natural group action dynamics on a family of cubic surfaces: real and complex
Time and place
12:30–1:30 PM on Thursday, September 21st, 2023; NAC 6/114
Roland Roeder (IUPUI)
Abstract
There is a beautiful dynamical system that one obtains by considering a certain family of cubic surfaces SD in three dimensional Euclidean Space ℝ3. There are three natural involutions of SD arising from intersections of SD with suitable lines parallel to the coordinate axes in ℝ3. While each involution has trivial dynamics, the group generated by taking arbitrary compositions of them has quite rich dynamics. In particular, for some parameters D, the surface SD has multiple connected components and the group action dynamics is ergodic (in particular transitive) on some components while properly discontinuous on other ones. This goes back to work of W. Goldman in 2003.
I will explain the above topic at a rather elementary level. I will then discuss what happens when one considers the corresponding complex cubic surface in ℂ3, which are connected for every choice of D. For suitable parameters D the transitive and properly discontinuous dynamics coexist on disjoint open subsets of the same complex surface. Several open questions follow. This later part of this talk is based on recent joint work with Julio Rebelo.
