[Paper Review] Zero divisors and units with small supports in group algebras of torsion-free groups

Aliréza Abdollahi, Zahra Taheri|arXiv (Cornell University)|Apr 29, 2017

Finite Group Theory Research14 references8 citations

TL;DR

This paper investigates zero divisors and units with small supports in group algebras of torsion-free groups, focusing on elements of support size 3. Using combinatorial graph-theoretic methods, particularly zero-divisor and unit graphs, the authors prove that if αβ = 0 with |supp(α)| = 3, then |supp(β)| ≥ 10 over any field, and |supp(β)| ≥ 20 over F₂; for units, if αβ = 1, then |supp(β)| ≥ 9 over any field, improving prior bounds and establishing forbidden subgraphs in the associated graphs.

ABSTRACT

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

Motivation & Objective

• To determine the minimal possible support size of a non-zero element β in F[G] such that αβ = 0 for some α ∈ F[G] with |supp(α)| = 3, where G is a torsion-free group and F is any field.
• To improve existing lower bounds on the support size of β in the zero divisor case, especially over the field F₂, and for the unit case where αβ = 1.
• To characterize forbidden subgraphs in zero-divisor and unit graphs associated with such elements, using combinatorial and graph-theoretic techniques.
• To establish that certain graph structures (e.g., triangles, K₂,₃, specific cycles) cannot appear as subgraphs in the zero-divisor or unit graphs of length-3 elements.
• To provide a structural understanding of the algebraic constraints imposed by small-support elements in group rings of torsion-free groups.

Proposed method

• Define the zero-divisor graph Z(α, β) for non-zero α, β ∈ F[G] with αβ = 0, using a combinatorial construction based on matched rectangles.
• Introduce the unit graph U(γ, δ) for elements with γδ = 1, and prove it is simple and loopless, with no C₃–C₃ or K₂,₃ subgraphs.
• Use graph-theoretic invariants such as vertex degrees, cycle structures, and forbidden subgraphs to derive contradictions when support sizes are too small.
• Apply case analysis on the size of the product set supp(γ)supp(δ) for unit pairs, considering possible partitions and constraints from the group algebra multiplication.
• Employ isomorphism and subgraph exclusion techniques to show that certain configurations (e.g., multiple triangles, high-degree vertices) lead to contradictions.
• Use induction and counting arguments on the number of pairs (i,j) such that hᵢgⱼ = hᵢ′gⱼ′ to bound the size of support sets in the product.

Experimental results

Research questions

• RQ1 What is the minimal possible size of the support of β ∈ F[G] such that αβ = 0 for some non-zero α ∈ F[G] with |supp(α)| = 3 and G torsion-free?
• RQ2 How does the minimal support size of β change when the base field is F₂, compared to arbitrary fields?
• RQ3 What are the structural constraints on the unit graph U(γ, δ) when |supp(γ)| = 3 and γδ = 1 in F[G] for a torsion-free group G?
• RQ4 Which specific graph structures (e.g., triangles, complete bipartite graphs) cannot appear as subgraphs in zero-divisor or unit graphs of length-3 elements?
• RQ5 Can the support size of β be bounded below using combinatorial arguments on the number of group elements in the product set supp(γ)supp(δ) when γδ = 1?

Key findings

• For any field F and torsion-free group G, if αβ = 0 with |supp(α)| = 3, then |supp(β)| ≥ 10.
• Over the field F₂, the lower bound improves to |supp(β)| ≥ 20 when αβ = 0 and |supp(α)| = 3.
• For units, if αβ = 1 with |supp(α)| = 3, then |supp(β)| ≥ 9 over any field, improving previous results.
• The zero-divisor graph Z(α, β) cannot contain any of the graphs in Figure 1 as a subgraph, regardless of the field or torsion-free group.
• Over F₂, the zero-divisor graph cannot contain any of the graphs listed in Table 1 as a subgraph.
• The unit graph U(γ, δ) is simple and contains no C₃–C₃ or K₂,₃ subgraphs, and no configuration with |supp(δ)| = 8 is possible without contradiction.