Structure of Boundaries of 3-Dimensional Convex Divisible Domains

Geometry Topology Student Seminar
Wednesday, April 3, 2024 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Alex Nolte – Georgia Tech
Thomas Rodewald

I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.