Some diophantine problems arising from the theory of cyclically-presented groups

R. W. K. Odoni

doi:10.1017/S0017089599950383

Abstract

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Let n∈ℕ and let Fn be the free group on n generators. Let w be an arbitrary word in Fn, and let σ be an n-cycle in Sn. We consider groups of the type Γ(n,w)=Fn/N, where N is the normal closure in Fn of the “cycled words’’ w, σ(w), σ2(w),…,σn−1(w), and solve, by means of classical algebraic number theory, the following problems.

A. When is Γ(n,w)ab infinite?

B. When is Γ(n,w) a perfect group?

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

CAVICCHIOLI, ALBERTO REPOVŠ, DUŠAN and SPAGGIARI, FULVIA 2003. TOPOLOGICAL PROPERTIES OF CYCLICALLY PRESENTED GROUPS. Journal of Knot Theory and Its Ramifications, Vol. 12, Issue. 02, p. 243.

Williams, Gerald 2009. The aspherical Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups. Journal of Group Theory, Vol. 12, Issue. 1,

CREMONA, J. E. and EDJVET, M. 2010. CYCLICALLY PRESENTED GROUPS AND RESULTANTS. International Journal of Algebra and Computation, Vol. 20, Issue. 03, p. 417.

WILLIAMS, GERALD 2010. UNIMODULAR INTEGER CIRCULANTS ASSOCIATED WITH TRINOMIALS. International Journal of Number Theory, Vol. 06, Issue. 04, p. 869.

WILLIAMS, GERALD 2012. GROUPS OF FIBONACCI TYPE REVISITED. International Journal of Algebra and Computation, Vol. 22, Issue. 08, p. 1240002.

Bogley, William A. and Williams, Gerald 2016. Efficient finite groups arising in the study of relative asphericity. Mathematische Zeitschrift, Vol. 284, Issue. 1-2, p. 507.

Bogley, William A. Edjvet, Martin and Williams, Gerald 2019. Groups St Andrews 2017 in Birmingham. p. 169.

Chinyere, Ihechukwu and Bainson, Bernard Oduoku 2021. Perfect Prishchepov groups. Journal of Algebra, Vol. 588, Issue. , p. 515.

Chinyere, Ihechukwu and Williams, Gerald 2021. Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups. Journal of Algebra, Vol. 580, Issue. , p. 104.

Noferini, Vanni and Williams, Gerald 2022. Cyclically presented groups as Labelled Oriented Graph groups. Journal of Algebra, Vol. 605, Issue. , p. 179.

Chinyere, Ihechukwu and Williams, Gerald 2022. Fractional Fibonacci groups with an odd number of generators. Topology and its Applications, Vol. 312, Issue. , p. 108083.

Mohamed, Esamaldeen and Williams, Gerald 2023. Counting isomorphism classes of groups of Fibonacci type with a prime power number of generators. Journal of Algebra, Vol. 633, Issue. , p. 887.

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Some diophantine problems arising from the theory of cyclically-presented groups

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