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.


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


Zbl 1013.20026
Full Text: DOI Euclid