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Growth series of stem products of cyclic groups. (English) Zbl 1013.20026

From the introduction: Let \(G\) be a group with a finite generating set \(X\) and let \(g\in G\). Then we define the length of \(g\) (with respect to \(X\)), denoted \(l(g)\), to be the least integer \(n\) such that \(g\) can be written as a product \(g=x_1\cdots x_n\), where \(x_i\in X\cup X^{-1}\) (\(1\leq i\leq n\)).
The growth series of \(G\) (with respect to \(X\)) is defined to be \(\gamma_G(t)=\sum^\infty_{n=0}a_nt^n\), where \(a_n\) denotes the number of elements of length \(n\). The group \(G\) is said to have rational growth (with respect to \(X\)) if \(\gamma_G(t)\) is a rational function.
There has recently been much interest in either calculating explicitly or proving rationality of the growth series for various groups (almost always with respect to specific generating sets), and it is our purpose here to add to the list of known examples.
In fact we are interested in the following stem products of cyclic groups: \[ G_r=\langle x_1,x_2,\dots,x_r\mid x^{p_1}_1=x^{p_2}_2=\cdots=x^{p_r}_r\rangle\quad (2\leq p_1\leq p_2\leq\cdots\leq p_r;\;r\geq 2) \] when the factors are infinite and when the factors are finite, \[ \begin{split} H_{r,s}=\langle x_1,x_2,\dots,x_r\mid x^{sp_1}_1=x^{sp_2}_2=\cdots=x^{sp_r}_r=1,\;x^{p_1}_1=x^{p_2}_2=\cdots=x^{p_r}_r\rangle\\ (2\leq p_1\leq p_2\leq\cdots\leq p_r;\;r\geq 2;\;s\geq 2).\end{split} \] We first consider the case when \(r=s=2\). That is, we study the groups \(G_2=\langle x,y\mid x^p=y^q\rangle\) and \(H_{2,2}=\langle x,y\mid x^{2p}=y^{2q}=1,\;x^p=y^q\rangle\), where \(2\leq p\leq q\).
In Section 2 we explicitly calculate the growth series of \(G_2\) with respect to \(\{x,y\}\). In Section 3 in Theorem 3.3.1 we give the growth series \(\gamma_{2,2}(t)\) of \(H_{2,2}\) with respect to \(\{x,y\}\).
In Section 4 we verify that if \(\chi(H_{2,2})\) denotes the Euler characteristic of \(H_{2,2}\) then \(\gamma_{2,2}(1)={1\over\chi(H_{2,2})}\). (The identity \(\gamma_G(1)={1\over\chi(G)}\) is a phenomenon that occurs for many examples of groups \(G\).) This supports a conjecture of M. Edjvet which we state and briefly discuss in Section 4.
Finally, in Section 5.1 we indicate how the results of Section 2 can be extended to prove that \(G_r\) has rational growth series with respect to \(\{x_1,\dots,x_r\}\) for \(r\geq 2\); and in Section 5.2 we extend the results of Section 3 to prove that \(H_{r,s}\) has rational growth series with respect to \(\{x_1,\dots,x_r\}\) for \(r\geq 2\) and \(s\geq 2\).

MSC:

20F05 Generators, relations, and presentations of groups
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[1] DOI: 10.1007/BF01200431 · Zbl 0820.20036 · doi:10.1007/BF01200431
[2] DOI: 10.1017/S1446788700035035 · doi:10.1017/S1446788700035035
[3] Grigorchuk R. I., Matematicheskie Zametki 58 (5) pp 653– (1995)
[4] Johnson D. L., Berlin pp 157– (1995)
[5] DOI: 10.1007/BF01388836 · Zbl 0577.20020 · doi:10.1007/BF01388836
[6] DOI: 10.1017/S030500410003499X · doi:10.1017/S030500410003499X
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