  We show that if a finite, large enough subset A of an arbitrary abelian group
satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|,
then A must contain a three-term arithmetic progression whose terms are not all
equal, and A + A must contain an arithmetic progression or a coset of a
subgroup, either of which of size at least exp^[ c (log |A|)^{delta} ]. This
extends analogous results obtained by Sanders and, respectively, by Croot, Laba
and Sisask in the case where the group is that of the integers or a finite
field.
