Essential expansion and Kazhdan Property (T)

Abstract: We say that a sequence of finite, d-regular graphs is
"essentially expander" if it can be turned into a disjoint union of
expanders after removing and adding o(n) edges. We give several
characterizations of such sequences. We solve Bowen's problem proving
that the local (Benjamini-Schramm) approximation of a finitely
generated group with Kazhdan Property (T) is essentially expander. We
use our characterization to answer Freedman's question about mapping
large subcomplexes of 2-dimensional simplicial complexes to
1-dimensional subcomplexes.