A non-Euclidean story or: how to persist when your geometry doesn't

Abstract

Too little mathematics has been written in prose. Thus we prove here, via a fantasy novellette, that a locally L-bilipschitz mapping $f \colon X \to Y$ between uniformly Ahlfors $q$-regular, complete and locally compact path-metric spaces $X$ and $Y$ is an $L$-bilipschitz map when $Y$ is simply connected. The motivation for such a result arises from studying the asymptotic values of BLD-mappings with an empty branch set; see e.g. [L17]. As far as the author is aware, the result is new, even though it would not be hard for specialists in the field to prove. The proof is essentially a modest extension of the ideas in [L17] in a more general setting when the branch set is empty.