Contractibility of Vietoris-Rips Complexes of dense subsets in $({\mathbb{R}}^n,{\mathrm{\ell }}_1)$ via hyperconvex embeddings

We consider the contractibility of Vietoris-Rips complexes of dense subsets of $({\mathbb{R}}^n,{\mathrm{\ell }}_1)$ with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each $n$ natural there is a $r_n>0$ so that the Vietoris-Rips complex of $({\mathbb{Z}}^n,{\mathrm{\ell }}_1)$ at scale $r$ is contractible for all $r\ge r_n$. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space $X$ and its connection to the Vietoris-Rips complex of $X$. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$.