On consecutive factors of the lower central series of right-angled Coxeter groups

TL;DR

This work advances the understanding of the lower central series of right-angled Coxeter groups by constructing a concrete Magnus embedding and deriving an explicit description of the fourth graded component $L^4(RC_K)$. It provides a complete classification for $K$ on four vertices, highlighting how the edge structure of $K^1$ governs the rank and explicit nested-commutator generators, and offers an algorithm to build $L^4(RC_K)$ from four-point subcomplexes for general $K$. Compared with the RA_K case, the results reveal richer behavior due to the involutive relations $g_i^2=1$, and the approach yields generators that are entirely nested commutators. The findings have implications for the algebraic structure of $RC_K$ and its associated Lie algebra, with potential applications to computations in combinatorial and geometric group theory.

Abstract

We study the lower central series of the right-angled Coxeter group $RC_\mathcal K$ and the corresponding associated graded Lie algebra $L(RC_\mathcal K)$ and describe the basis of the fourth graded component of $L(RC_\mathcal K)$ for any $\mathcal K$.

Theorems & Definitions (32)

• Theorem 2.1: pa-ve
• Theorem 2.2: bu-pa00, bu-pa15
• Theorem 2.3: pa-ve
• Remark
• Corollary 2.4
• Proposition 2.5
• Proposition 2.6: ve3
• Proposition 2.7: WaldingerLCS
• Proposition 2.8: WaldingerLCS
• Proposition 2.9: WaldingerLCS