Quasisymmetrically co-Hopfian Sierpiński Spaces and Menger Curve

Abstract

A metric space $X$ is quasisymmetrically co-Hopfian if every quasisymmetric embedding of $X$ into itself is onto. We construct the first examples of metric spaces homeomorphic to the universal Menger curve and higher dimensional Sierpiński spaces, which are quasisymmetrically co-Hopfian. We also show that the collection of quasisymmetric equivalence classes of spaces homeomorphic to the Menger curve is uncountable. These results answer a problem and generalize results of Merenkov from \cite{Mer:coHopf}.

63 pages. A new result has been added, showing that the collection of quasisymmetric equivalence classes of Menger curves is uncountable

References in corpus (1)

• No-splitting property and boundaries of random groups