## 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

Zbl 1013.20026
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