A note on asymptotic cones of graph-adapted smocked spaces

Abstract

Smocked spaces, introduced by Sormani and collaborators as a generalization of pulled thread spaces, provide a broad class of metric quotients of Euclidean space. In this note we investigate their large-scale geometry via periodic graph models. We introduce the notion of a graph-adapted smocked realization of a periodic weighted graph and establish a uniform additive distortion estimate between the smocked metric and the underlying graph metric. As a consequence, graph-adapted smocking preserves stable norms and asymptotic cones. Combining this with classical homogenization results for periodic graphs, we show that the tangent cone at infinity of a graph-adapted smocked space is determined by the stable norm of the associated periodic graph, which implies that every centrally symmetric rational polyhedral norm arises as the unique tangent cone at infinity of a graph-adapted smocked space. This establishes a connection between stable norm theory and the asymptotic geometry of smocked spaces.