Seminar

In the Riemannian symmetric space X = SL(3,R)/SO(3), consider a totally geodesic hyperbolic plane Y arising as an SO(2,1)-orbit. Given a geodesic L contained in Y, the higher-rank geometry allows one to translate Y along a transverse maximal flat, producing a one-parameter family of “floating” geodesic planes Y_{L,t}.

 

In this talk, I will describe the geometric mechanism behind the following result: as the height parameter t tends to infinity, the nearest-point projection of Y_{L,t} onto the reference plane Y converges to the geodesic L. This is a key ingredient in the construction of floating geodesic planes whose projections to quotients of X by Zariski-dense discrete subgroups of SL(3,R) have fractal closures.

A distinctive feature of the picture is that the fibers of the nearest-point projection contain entire maximal flats, leading to parallel families of geodesics and more intricate asymptotic configurations. The proof combines a detailed analysis of Busemann functions and the interaction with the visual boundary of the symmetric space.