On the correspondence between biLipschitz mappings and quasisimilarities on cylinders

Given a complete bounded metric space ( 𝑍 , 𝑑 𝑍 ) whose diameter D is greater than 0, one can define its cylinder by C o n ⁡ ( 𝑍 ) : = 𝑍 × ( 0 , 𝐷 ] . This paper studies the correspondence between mapping classes on hyperbolic cones and on ℓ 1 -metric cylinders, obtained by endowing cylinders with hyperbolic and ℓ 1 -metrics respectively. The main result establishes a one-to-one correspondence between the biLipschitz classes of hyperbolic cones and the quasisimilarity classes of ℓ 1 -metric cylinders. This extends the analogue of [2, Theorem 1.1] to a class of Gromov hyperbolic spaces that are not necessarily proper or connected.