  In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a
Banach space that is a uniform quotient of $\ell_p$, $1 < p \neq 2 < \infty$,
must be isomorphic to a linear quotient of $\ell_p$. 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.
