The spherical boundary and volume growth (Q415748)

Summary: We consider the \textit{spherical boundary}, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is ``small'' (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies to \(l^2\)-products of length spaces, since we prove that a natural metric, generalizing such ``norm-like'' product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.