An asphericity conjecture and Kaplansky problem on zero divisors

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Suppose a group representation H = 〈A∥ℛ〉 is aspherical, nx ∉ A, W(A ∪ x) is a word in alphabet (A ∪ x)±1 with nonzero sum of exponents on x, and the group H naturally embeds in G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉. It is conjectured that the presentation G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉 is aspherical if and only if G is torsion free. It is proven that if this conecture is false and G = 〈A ∪ x ∥ℛ ∪ W(A ∪ x)〉 is a counter example, then the integral group ring Z(G) of torsion free group G will contain zero divisors. Some special cases when this conjecture holds are also indicated.

Original languageEnglish (US)
Pages (from-to)13-19
Number of pages7
JournalJournal of Algebra
Issue number1
StatePublished - Jun 1 1999

ASJC Scopus subject areas

  • Algebra and Number Theory


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