Property (β) and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of a"" (p) , 1 < p not equal 2 < a, must be isomorphic to a linear quotient of a"" (p) . We apply the geometric property (beta) of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case 1 < p < 2. We also give a necessary condition for a Banach space to have c (0) as a uniform quotient.