  We discuss a problem posed by Gersten: Is every automatic group which does
not contain Z+Z subgroup, hyperbolic? To study this question, we define the
notion of "n-tracks of length n", which is a structure like Z+Z, and prove its
existence in the non-hyperbolic automatic groups with mild conditions. As an
application, we show that if a group acts effectively, cellularly, properly
discontinuously and cocompactly on a CAT(0) cube complex and its quotient is
"weakly special", then the above question is answered affirmatively.
