Quasisymmetrically co-Hopfian Menger curves and Sierpiński spaces

Abstract

In this paper we introduce and study slit Sierpiński spaces and slit Menger curves. These metric spaces appear as limits of slit domains in Rn,n≥2, equipped with the inner metric. Using these slit spaces we construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically (QS) co-Hopfian. Here a metric space X is QS co-Hopfian if every quasisymmetric embedding of X into itself is onto. It is also shown that the collection of quasisymmetric equivalence classes of slit Menger curves is uncountable. These results answer a problem and generalize results of Merenkov from [34].