On intersecting subgroups of Brunnian link groups

Abstract

Let $G\left(L_n\right)$ be the link group of a Brunnian $n$–link $L_n$ and $R_i$ be the normal closure of the $i^{th}$ meridian in $G\left(L_n\right)$ for $1\le i\le n$. In this article, we show that the intersecting subgroup $R_1\cap R_2\cap \cdots \cap R_m$ coincides with the iterated symmetric commutator subgroup ${\prod }_{\sigma \in {\Sigma }_m}\left[\left[R_{\sigma \left(1\right)},R_{\sigma \left(2\right)}\right],\dots ,R_{\sigma \left(m\right)}\right]$ for $2\le m\le n$ using the techniques of homotopy theory. Moreover, we give a presentation for the intersecting subgroup $R_1\cap R_2\cap \cdots \cap R_n$.