Finite quotients of Bruhat-Tits buildings as geometric expanders

In cite{FGLNP}, Fox, Gromov, Lafforgue, Naor and Pach, in a respond to a question of Gromov cite{G}, constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of $tilde{A_d}$-buildings, for $dgeq 2$, over local fields. In this paper, this result is extended to general high rank Bruhat-Tits buildings.

Mixing properties and the chromatic number of Ramanujan complexes

Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have high girth and high chromatic number, generalizing a well known result about Ramanujan graphs.