A relative Sierpinski theorem: Erratum to “Nonhyperbolic Coxeter groups with Menger boundary”

Abstract

The purpose of this erratum is to correct the proof of Proposition 2.3 of [HHS]. A classical theorem of Sierpinski states that every subspace of dimension at most one in the 2-dimensional disc $D^2$ can be topologically embedded in the Sierpinski carpet. The proof of Proposition 2.3 of [HHS] implicitly provides a relative version of Sierpinski’s theorem. Unfortunately the proof of Proposition 2.3 given in [HHS] is incorrect. We provide two brief proofs that each fill this gap. One is a self-contained argument suited specifically for the needs of [HHS], and in the other we explicitly prove a relative embedding theorem that produces embeddings in the Sierpinski carpet with certain prescribed boundary values.

Link to the paper corrected hereby.