Strictly ascending HNN extensions in soluble groups. (English) Zbl 1305.20041
Summary: We show that there exist finitely generated soluble groups which are not LERF but which do not contain strictly ascending HNN extensions of a cyclic group. This solves Problem 16.2 in the Kourovka notebook (2006; Zbl 1084.20001). We further show that there is a finitely presented soluble group which is not LERF but which does not contain a strictly ascending HNN extension of a polycyclic group.
MSC:
| 20F16 | Solvable groups, supersolvable groups |
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20E26 | Residual properties and generalizations; residually finite groups |
| 20E07 | Subgroup theorems; subgroup growth |
| 20F05 | Generators, relations, and presentations of groups |
Keywords:
finitely generated soluble groups; LERF groups; strictly ascending HNN extensions; finitely presented soluble groups; polycyclic groupsCitations:
Zbl 1084.20001References:
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