The transfer of property $(\beta )$ of Rolewicz by a uniform quotient map

We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(\beta )$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(\beta )$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(\beta )$-moduli of the same power-type even under renorming.