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| one-relator group | |
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| Name | One-relator group |
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one-relator group A one-relator group is a group defined by a presentation with a single defining relation, typically written <X | r>. Origins of the subject trace to work of Wilhelm Magnus, Max Dehn, and developments in the Combinatorial group theory tradition; the concept plays a central role in studies of knot theory, 3-manifold groups, and algorithmic problems in algebra. One-relator groups provide fertile connections to groups such as free groups, surface groups, and examples arising from HNN extensions and amalgamated products.
A one-relator group is given by a finite generating set X and a single relator r, written <X | r>, where r is a cyclically reduced word in the free group on X. Classical examples include the Baumslag–Solitar groups such as BS(1,n), fundamental groups of closed orientable surfaces like the genus g surface group, and knot groups such as the trefoil knot group and the figure-eight knot group. Early research by Max Dehn and Wilhelm Magnus produced canonical examples including torsion-free one-relator groups and one-relator groups with torsion studied by O. Schreier and Heinz Prüfer-type constructions.
The Freiheitssatz, proved by Wilhelm Magnus, asserts that in a one-relator presentation <X | r> where r involves a subset of generators, the subgroup generated by the remaining generators is free; this result links to the Nielsen–Schreier theorem and underpins structural analyses by John Stallings and Gersten. Consequences include injectivity results for inclusion maps of subgroups and decomposition theorems applied by researchers such as Hyman Bass and Gilbert Baumslag. The Freiheitssatz also plays a role in proofs of residual properties studied by Graham Higman and results about cohomological dimensions related to work of C. T. C. Wall.
One-relator groups exhibit diverse algebraic and geometric properties: torsion behavior analyzed by M. J. Dunwoody and J. Howie, cohomological dimension influenced by Kenneth S. Brown's work, and subgroup separability investigated by Peter Scott and D. Wise. The structure of centralizers and normalizers in one-relator groups has connections to results of Magnus and Baumslag, while growth and amenability relate to studies by Mikhail Gromov and I. G. Macdonald. Invariants such as the deficiency, Euler characteristic, and Bieri–Neumann–Strebel invariants have been computed in cases studied by Robert Bieri and Ralf Strebel; homological properties connect to the L^2 Betti number theory developed by Lück.
Algorithmic aspects include solvability of the word problem, conjugacy problem, and isomorphism problem for many classes of one-relator groups. Magnus provided an algorithmic solution to the word problem for one-relator groups, influencing later algorithmic work by G. Higman, Donald Solitar, and Gerald A. Miller. Conjugacy problems were treated by W. Magnus and refined by R. Lyndon and P. Schupp, while complexity and decidability boundaries have been explored in the context of Baumslag–Solitar groups and constructions by Gilbert Baumslag and Bridson. Connections to Turing machine simulations and undecidability results have been studied by Martin Higman and Pierre Deligne in broader combinatorial settings.
One-relator groups with torsion (where r is a proper power) were investigated by F. C. R. Spelling and later by J. Howie; results include asphericity and subgroup structure theorems by B. Bogley and S. Pride. Surface groups (fundamental groups of closed surfaces) are classical one-relator examples studied by Henri Poincaré and Max Dehn with deep connections to Teichmüller theory and Thurston’s work on 3-manifolds. Limit groups, as developed by Zlil Sela, O. Kharlampovich, and A. Myasnikov, appear as limits of free groups and share intersectional phenomena with one-relator groups, informing solutions to equations over groups and Tarski problems.
Constructions include forming one-relator presentations via Tietze transformations studied by Heinz Tietze and using HNN extensions introduced by H. Neumann and G. Higman to build complex examples; amalgamated free products studied by Kurosh provide ways to glue one-relator groups while preserving or altering properties. Techniques from small cancellation theory by Roger Lyndon and Paul Schupp are applied to control geometric features; JSJ decompositions used by Sela and Rips connect one-relator groups to splittings over cyclic subgroups studied by Dunwoody and Scott–Swarup decompositions.