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Presentations of the amalgamated free product of two infinite cycles

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Abstract

LetH=〈a,b;a k=b l〉, wherek,l≧2 andk+l>4. McCool and Pietrowski have proved that any pair of generators forH is Nielsen equivalent to a pairx=a r andy=b s where

$$(a){\text{ }}gcd(r, s) = gcd(r, k) = gcd(s, l) = 1,$$
$$(b){\text{ }}0< 2r \leqq ks{\text{ }}and{\text{ }}0< 2s \leqq lr.$$

In terms ofx andy,H can be presented as

$$G = \left\langle {x,{\text{ }}y;{\text{ }}x^{ks} = y^{lr} ,\left[ {x,{\text{ }}y^l } \right] = \left[ {x^k ,{\text{ }}y} \right] = 1} \right\rangle$$

and Zieschang has shown that ifr=1 ors=1, thenH can be defined by a single relation inx andy.

We establish the exact converse of Zieschang's result, namely thatH is not defined by a single relation inx andy unlessr=1 ors=1. The proof is based on an observation of Magnus which associates polynomials with relators and some elementary facts about cyclotomic polynomials.

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Authors and Affiliations

  1. Department of Pure Mathematics, Queen Mary College, Mile End Road, El 4NS, London, United Kingdom

    Donald J. Collins

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  1. Donald J. Collins
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Collins, D.J. Presentations of the amalgamated free product of two infinite cycles. Math. Ann. 237, 233–241 (1978). https://doi.org/10.1007/BF01420178

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