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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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