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On the dependence of the growth rate on the length of the defining relator

Authors:	A G Shukhov

Institution:	(1) Moscow Institute of Radio Engineering, Electronics, and Automation, Moscow, USSR

Abstract:	Let { $G_k = \left\langle {a_1 ,...,a_m \left\| {r_1^{\left\{ k \right\}} = ... = r_{jk}^{(k)} = 1} \right.} \right\rangle = \left\langle {A\left\| {R_k } \right.} \right\rangle$ } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.

Keywords:	sequence of finitely presented groups growth rate small cancellation condition Greendlinger’ s lemma one-relator group
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