Natural group action dynamics on a family of cubic surfaces: real and complex

Roland Roeder (IUPUI)

Abstract

There is a beautiful dynamical system that one obtains by considering a certain family of cubic surfaces S_D in three-dimensional Euclidean Space ℝ³. There are three natural involutions of S_D arising from intersections of S_D 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 S_D 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.