  In this paper, the first of a series of two, we continue the study of higher
index theory for expanders. We prove that if a sequence of graphs is an
expander and the girth of the graphs tends to infinity, then the coarse
Baum-Connes assembly map is injective, but not surjective, for the associated
metric space $X$.
  Expanders with this girth property are a necessary ingredient in the
construction of the so-called `Gromov monster' groups that (coarsely) contain
expanders in their Cayley graphs. We use this connection to show that the
Baum-Connes assembly map with certain coefficients is injective but not
surjective for these groups. Using the results of the second paper in this
series, we also show that the maximal Baum-Cones assembly map with these
coefficients is an isomorphism.
