Geometric property (T) and sofic approximations

It is a classical result that one can obtain a sequence of expander graphs from a residually finite group with property (T), e.g. $SL(n,\mathbb{Z})$ for $n\ge 3$. Willet and Yu showed that these sequences of graphs even satisfy the stronger notion of geometric property (T), a property recently introduced by them to construct counterexamples to the coarse Baum-Connes conjecture. Going from residually finite groups to the more general sofic groups, another recent result by Kun says that sofic approximations of property (T) groups are up to a vanishing proportion of error terms a union of expander graphs. Building further on work of Alekseev and Finn-Sell and on work of Winkel we prove that sofic property (T) groups always admit a sofic approximation satisfying geometric property (T), which also characterizes groups with property (T) among the sofic groups. The tools used in the proof also apply to the case of general bounded degree graph sequences away from groups.