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A new formula for the spherical growth series of an amalgamated free product of two infinite cyclic groups. (English) Zbl 1436.20041

Let \(G(p,q)\) be the group with presentation \(\langle x, y \mid x^{p} = y^{q} \rangle\) where \(p\), \(q\) are integers such that \(2 \leq p \leq q\).
In the paper under review, an explicit formula for the spherical growth series of the group \(G(p,q)\) with respect to the generating set \(\{x,y,x^{-1},y^{-1}\}\) is obtained. The author remark that a similar formula has been previously obtained by C. P. Gill [Int. J. Algebra Comput. 9, No. 1, 1–30 (1999; Zbl 1013.20026)]) but the new formula is deduced with independent methods and can be used to compute explicitly the growth series as rational function.

MSC:

20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F69 Asymptotic properties of groups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory

Citations:

Zbl 1013.20026
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