Uniform Spectral Gaps for SL(2,Z)Z^2 and Kazhdan’s Property (T).

In 1973, Margulis gave the first explicit construction of expander graphs, exploiting a property introduced by Kazhdan in 1967. In subsequent years, the links between amenability, Kazhdan’s Property (T), and paradoxical decompositions have been greatly clarified. Since then, many results have been proven, most notably thanks to the “Bourgain-Gamburd machine” (2005) and arithmetic combinatorics; this gave new life to old open problems and conjectures on group expansion. In this talk, we revisit Margulis’ first construction using the group SL(2,Z) x Z², tell the current story by surveying recent results first obtained by Bourgain and Gamburd, followed by several notable results, and present some new results on uniform spectral gaps for unitary representations of SL(2,Z) x Z².