Sublinear Rigidity of Lattices in Semisimple Lie Groups

I will discuss metric deformations of lattices in semisimple Lie groups. The underlying question is whether the class of lattices is stable (rigid) under distortions of 'sublinear' nature. I will present work that settles this question in the affirmative in almost all settings. For groups without real rank 1 factors, this amounts to a sublinear generalization of the classical quasi-isometric rigidity results.
The main focus of my talk will be the geometric structure of non-uniform lattices and its relation to the horospheres of the corresponding symmetric space. I aim to describe the proof of a key proposition, which is motivated by a lattice criterion conjectured by Margulis and proven by Oh and Benoist-Miquel.