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In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether aBanach 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 geometricproperty $(\beta)$ of Rolewicz to the study of uniform and Lipschitz quotientmaps, and answer the above question positively for the case $1<p<2$. We alsogive a necessary condition for a Banach space to have $c_0$ as a uniformquotient.

Property $(\beta)$ and uniform quotient maps