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On zero divisors with small support in group rings of torsion-free groups. (English) Zbl 1292.20007

Author’s summary: Kaplansky’s zero divisor conjecture envisions that for a torsion-free group \(G\) and an integral domain \(R\), the group ring \(R[G]\) does not contain non-trivial zero divisors. We define the length of an element \(\alpha\in R[G]\) as the minimal non-negative integer \(k\) for which there are ring elements \(r_1,\ldots,r_k\in R\) and group elements \(g_1,\ldots,g_k\in G\) such that \(\alpha=r_1g_1+\cdots+r_kg_k\). We investigate the conjecture when \(R\) is the field of rational numbers. By a reduction to the finite field with two elements, we show that if \(\alpha\beta=0\) for non-trivial elements in the group ring of a torsion-free group over the rationals, then the lengths of \(\alpha\) and \(\beta\) cannot be among certain combinations. More precisely, we show for various pairs of integers \((i,j)\) that if one of the lengths is at most \(i\), then the other length must exceed \(j\). Using combinatorial arguments we show this for the pairs \((3,6)\) and \((4,4)\). With a computer-assisted approach we strengthen this to show the statement holds for the pairs and \((4,7)\). As part of our method, we describe a combinatorial structure, which we call matched rectangles, and show that for these a canonical labeling can be computed in quadratic time. Each matched rectangle gives rise to a presentation of a group. These associated groups are universal in the sense that there is no counter-example to the conjecture among them if and only if the conjecture is true over the rationals.

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
16Z05 Computational aspects of associative rings (general theory)
68W30 Symbolic computation and algebraic computation

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