The Bieri–Strebel Theorems

  • Katalin A. Bencsáth
  • Marianna C. Bonanome
  • Margaret H. Dean
  • Marcos Zyman
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

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.

Keywords

Bieri–Strebel theorem Finitely presented solvable group Subgroup of finite index Ascending HNN-extension Infinite cyclic quotient Solvable series 

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Copyright information

© Katalin A. Bencsath, Marianna C. Bonanome, Margaret H. Dean, Marcos Zyman 2013

Authors and Affiliations

  • Katalin A. Bencsáth
    • 1
  • Marianna C. Bonanome
    • 2
  • Margaret H. Dean
    • 3
  • Marcos Zyman
    • 3
  1. 1.Department of Mathematics and Computer ScienceManhattan CollegeNew YorkUSA
  2. 2.Department of Applied Mathematics and Computer Science New York City College of TechnologyThe City University of New YorkBrooklynUSA
  3. 3.Department of Mathematics Borough of Manhattan Community CollegeThe City University of New YorkNew YorkUSA

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