Paper Info
Title
On Zero Divisors with Small Support in Group Rings of Torsion-Free Groups
Abstract
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 is an element of R[G] as the minimal non-negative integer k for which there are ring elements r(1),..., r(k) is an element of R and group elements g(1),..., g(k) is an element of G such that alpha = r(1)g(1) + ... + r(k) g(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 (3, 16) 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.
Year
DOI
Venue
2012
10.1515/jgt-2013-0017
JOURNAL OF GROUP THEORY
Field
DocType
Volume
Integer,Discrete mathematics,Combinatorics,Rational number,Finite field,Algebra,Integral domain,Group ring,Zero divisor,Counterexample,Presentation of a group,Mathematics
Journal
16
Issue
ISSN
Citations 
5
1433-5883
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Pascal Schweitzer1346.39