Conjugation-free groups, lower central series and line arrangements

2013/05/28 by Michaël Friedman, Friedman, Michael
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1305.6503

openalex publication_date 2013/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The quotients G_k/G_k+1 of the lower central series of a finitely presented group G are an important invariant of this group. In this work we investigate the ranks of these quotients in the case of a certain class of conjugation-free groups, which are groups generated by x₁,...,x_n, and having only cyclic relations: x_it x_i_t-1 ... x_i1 = x_i_t-1 ... x_i1 x_it = ... = x_i1 x_it ... x_i2. Using tools from group theory and from the theory of line arrangements we explicitly find these ranks, which depend only at the number and length of these cyclic relations. It follows that for these groups the associated graded Lie algebra gr(G) decomposes, in any degree, as a direct product of local components.

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