### Structure of Boundaries of 3-Dimensional Convex Divisible Domains

- Series
- Geometry Topology Student Seminar
- Time
- Wednesday, April 3, 2024 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Alex Nolte – Georgia Tech

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.