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A Grushko theorem for 1-acylindrical splittings. (English) Zbl 0987.20010
Let \(G=A*_CB\) be the free product with amalgamated subgroup \(C\). Assume that \(C\) is malnormal, that is, \(gCg^{-1}\cap C=\{1\}\) for all \(g\in G\setminus C\). It is known that the formula \[ \text{rank }G\geq\text{rank }A+\text{rank }B-\text{rank }C \] fails in general.
For instance, if \(G\) is given with \(A=\langle s_1,s_2,s_3\mid s^2_1=s^2_2=s^2_3=1\rangle\), \(B=\langle s_4,s_5,s_6\mid s^2_4=s^2_5=s^2_6=1\rangle\) and \(C=\langle s_1s_2s_3\rangle=\langle(s_4s_5s_6)^{-1}\rangle\cong\mathbb{Z}\) then \(C\) is malnormal, \(\text{rank }A=\text{rank }B=3\) and \(\text{rank }C=1\) but \(\text{rank }G=4\).
Here, the author shows the remarkable fact that \[ \text{rank }G\geq\tfrac 13(\text{rank }A+\text{rank }B-2\text{rank }C+5). \]

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E34 General structure theorems for groups
20F05 Generators, relations, and presentations of groups
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