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Recently, Lima and Randrianarivony pointed out the role of the property$(\beta)$ of Rolewicz in nonlinear quotient problems, and answered aten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman.In the present paper, we prove that the modulus of asymptotic uniformsmoothness of the range space of a uniform quotient map can be compared withthe modulus of $(\beta)$ of the domain space. We also provide conditions underwhich this comparison can be improved.

Asymptotic geometry of Banach spaces and uniform quotient maps