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| finitely presented group | |
|---|---|
| Name | Finitely presented group |
| Type | Algebraic object |
| Field | Group theory, Algebraic topology |
| Introduced | 19th century |
| Notable | Max Dehn, William Burnside, Emil Artin, Otto Schreier, Alfred Tarski |
finitely presented group
A finitely presented group is a group described by a finite set of generators together with a finite set of relations among those generators, often given by a presentation. Such groups sit at the crossroads of Max Dehn's decision problems, William Burnside's problems on periodic groups, and constructions from Emil Artin and Otto Schreier, and they underpin many developments in Algebraic topology, Geometric group theory, and Combinatorial group theory.
A finitely presented group is specified by a finite generating set and a finite set of relators; classical notation uses a presentation G = ⟨S | R⟩ with S finite and R finite. The formalism originates in the work of Max Dehn on the word and conjugacy problems and was developed further by Otto Schreier and others in the context of combinatorial methods. Presentations allow explicit encodings that connect to decision problems studied by Emil Artin and later by logicians such as Alfred Tarski.
Common examples include the free group of finite rank (e.g., rank two free group considered by Emil Artin), finitely generated abelian groups like Cyclic groups and finite direct sums studied by William Burnside, and fundamental groups of compact surfaces as in the work of Henri Poincaré and Jakob Nielsen. Important families given by finite presentations include Baumslag–Solitar groups associated to Graham Higman's investigations, one-relator groups analyzed by W. Magnus, and Coxeter groups related to H.S.M. Coxeter and Branko Grünbaum. Non-examples arising in algorithmic contexts include certain recursively presented but not finitely presented groups appearing in constructions of Pyotr Novikov and Sergei Adian, and groups obtained by infinite amalgamations used by Jean-Pierre Serre.
Standard constructions producing finitely presented groups include quotients of finitely generated free groups (classical in Combinatorial group theory), HNN extensions introduced by G. Higman and further studied by H. Neumann, free products with amalgamation prominent in Jean-Pierre Serre's Bass–Serre theory, and direct or semidirect products used in constructions by William Burnside and R. L. Moore. Properties often explored are residual finiteness as in work by Malcev, coherence considered by Daniel Wise, and Hopficity linked to results of G. Baumslag. Finiteness properties FP_n and type F_n arise in the study of finiteness conditions used by K. S. Brown and C. T. C. Wall and relate to Eilenberg–MacLane spaces studied by Saunders Mac Lane.
The classic decision problems—word problem, conjugacy problem, and isomorphism problem—trace to Max Dehn and were central to the negative results by Pyotr Novikov and Sergei Adian who showed unsolvability in general. The Boone–Novikov constructions involve finitely presented groups encoding Turing machines, connecting to work by William Boone, Alonzo Church, and Alan Turing. Positive algorithmic results occur in special classes: one-relator groups studied by W. Magnus and hyperbolic groups defined by Mikhail Gromov have solvable word and conjugacy problems per contributions from Gromov and G. N. Arzhantseva. The isomorphism problem for finitely presented groups remains undecidable in general following proofs by G. Higman and extensions by G. Baumslag and V. A. Roman'kov.
Homological finiteness invariants such as homological type FP_n and cohomological dimension were developed in the work of K. S. Brown, C. T. C. Wall, and J. H. C. Whitehead, connecting group presentations to CW complexes and Eilenberg–MacLane spaces linked to Henri Poincaré's foundational ideas. Geometric group theory, advanced by Mikhail Gromov, studies finitely presented groups via Cayley graphs, quasi-isometries, and hyperbolicity relating to spaces studied by William Thurston and Michael Freedman. Boundaries at infinity, JSJ decompositions tied to William Jaco and Peter Shalen, and actions on CAT(0) spaces investigated by Bridson and Haefliger give geometric structure reflecting finite presentations.
Finitely presented groups provide explicit test cases and counterexamples shaping modern group theory, as in the negative solutions to classical decision problems by Pyotr Novikov and William Boone, and in constructions of exotic examples by R. Thompson and Graham Higman. They bridge group theory with Algebraic topology via fundamental groups of finite CW complexes studied by H. Hopf and Jean-Pierre Serre, inform low-dimensional topology through knot groups examined by J. W. Alexander and John Milnor, and underpin connections to logic exemplified in work by Alfred Tarski and Alan Turing. The interplay with geometric methods of Mikhail Gromov and structural decompositions of Jean-Pierre Serre continues to guide research in group theoretic classification and rigidity.