Abstracts for Mini-Workshop on "Geometric Group Theory in Bonn", January 31 - February 1, 2019

Cross ratios naturally arise on boundaries of negatively curved spaces
and are a valuable tool in their study. If one however slightly relaxes the
curvature assumption, simply requiring it to be *non-positive*, things
tend to get more complicated. Even the mere definition of a cross ratio becomes a
more delicate matter.

Restricting to the context of CAT(0) cube complexes $X$, we observe that
most issues disappear if one considers the $\ell^1$ metric on $X$, rather than the
CAT(0) metric. We obtain a canonical cross ratio on the horoboundary of the $\ell^1$

I will present a recent proof, by Kaluba, Nowak, and myself, of the fact that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n>4$.  I will discuss how the same strategy gives a new proof of property (T) for $\mathrm{SL}_n(\mathbb Z)$.