Representation theory and Ramanujan complexes

Expanders are sparse, highly connected graphs, and Ramanujan graphs are (in some sense) optimal expanders. They were first constructed using the automorphic representation theory of $GL_2$ by Lubotzky-Phillips-Sarnak and Margulis. The story gets considerably more complicated in higher dimensions, and is currently at the focus of extensive research. I will tell a part of this story, focusing on what do representations of $GL_3$ tell us on the topology and combinatorics of the so-called Ramanujan complexes.