2025/10/11 by Gabe Cunningham, Cunningham, Gabe, Igor Minevich +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Advanced Operator Algebra Research #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2510.10314
For a graph Γ and group G, GΓ is the subgroup of G|Γ| generated by elements with g in the coordinates corresponding to v and its neighbors in Γ. There is a natural epimorphism GΓ→ (G/[G,G])Γ with kernel [G,G]n ∩ GΓ. When [G,G]n ≤ GΓ, the structure of GΓ is easily described from (G/[G,G])Γ. Fixing Γ, if [G,G]|Γ| ≤ GΓ for all G, we say that Γ is RA (reducible to abelian). We showed in [2] that wide classes of graphs are RA, including graphs of girth 5 or more. The key tool is the RA matrix CΓ, and we showed that Γ is RA if and only if the row space Row(CΓ) = \mathbb Z|Γ|. Here, we study the possibilities for the elementary divisors of CΓ; the more nontrivial elementary divisors we get, the further Γ is from being RA (and the harder GΓ is to describe). We show that while many graphs, including those of girth 4, cartesian products, and most tensor products have at most one nontrivial elementary divisor, one can construct a graph of girth 3 with any prescribed set of elementary divisors and \mathbb Z-nullity.