The Bieri–Strebel Theorems
Proofs of two theorems on finitely generated groups by Bieri and Strebel are included in detail. The first theorem gives a dichotomy for finitely presented groups with an infinite cyclic quotient: such a group is either an ascending HNN-extension or else contains a free group of rank 2. An immediate consequence of the second theorem is that a solvable finitely presented group is either finite or else is virtually an ascending HNN-extension of a finitely generated solvable group.
KeywordsBieri–Strebel theorem Finitely presented solvable group Subgroup of finite index Ascending HNN-extension Infinite cyclic quotient Solvable series
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