Ramanujan graphs are k-regular graphs with all nontrivial eigenvalues are bounded ( in absolute value) by 2SR(k-1). Theyare optimal expanders (from spectral point of view). Explicit constructions ofsuch graphs were given in the 80's as quotients of the Bruhat-Tits treeassociated with GL(2) over a local field F, by the action of suitablecongruence subgroups of arithmetic groups.
Thespectral bound was proved using works of Hecke, Deligne and Drinfeld on the"Ramanujan conjecture" in the theory of automorphic forms.
The work of Lafforgue, extending Drinfeld from GL(2) to GL(n),opened the door for the construction of Ramanujan complexes as quotients of theBruhat-Tits buildings associated with GL(n) over F.
This way onegets finite simplical complxes which on one hand are "random like ''and at the same time have strong symmetries. These seemingly contradictingproperties make them very useful for constructions of various external objects.
Various applications have been found in combinatorics, coding theory and inrelation to Gromov's overlapping properties.
We will surveysome of these applications.