Cyclic Presentations in Algebraic Group Theory

Cyclic presentations constitute a class of group definitions in which the number of generators equals the number of relators and a cyclic permutation of the generators carries relators into one another. This symmetry endows these groups with a rich combinatorial structure, linking them to diverse areas such as geometric group theory, low‐dimensional topology and the theory of buildings. Central to their study is the star graph, a combinatorial object that encodes how generators and relators interact. Properties such as small cancellation conditions and asphericity can be read off from this graph, allowing one to determine whether the group is hyperbolic, SQ‐universal or satisfies the Tits alternative. Prominent families of cyclically presented groups include the Fibonacci, Sieradski and Gilbert–Howie groups, which serve both as test cases for new methods and as sources of exotic examples: for instance, they can yield non‐amenable torsion‐by‐cyclic groups or groups acting on Euclidean and hyperbolic buildings. Recent advances have refined the classification of presentations that give rise to particularly well behaved or large groups, illuminated torsion phenomena in fractional Fibonacci groups, and extended curvature‐distribution techniques to prove hyperbolicity in broad infinite families. Together these developments underscore the central role of cyclic symmetry in organising the landscape of finitely presented groups and in linking algebraic, geometric and computational perspectives.

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Cyclic Presentations in Algebraic Group Theory publication trend

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Technical terms

Cyclic presentation: A group presentation with as many generators as relators admitting a cyclic permutation symmetry among generators and relators.

Star graph: A bipartite graph encoding incidences between generators and relators in a cyclic presentation, used to detect small cancellation and geometric properties.

Small cancellation: A combinatorial condition on overlapped subwords of relators that guarantees geometric finiteness and often hyperbolicity.

Hyperbolicity: A property of groups exhibiting negative‐curvature behaviour in their Cayley graph, leading to strong algorithmic and structural consequences.

Tits alternative: A dichotomy stating that a finitely generated group either contains a non-abelian free subgroup or is virtually solvable.

References

1. Fractional Fibonacci groups with an odd number of generators. Topology and its Applications (2022).
2. Generalized polygons and star graphs of cyclic presentations of groups. Journal of Combinatorial Theory Series A (2022).
3. Redundant relators in cyclic presentations of groups. Journal of Group Theory (2023).
4. Curvature distribution and hyperbolicity. Journal of Group Theory (2023).
5. Non-amenable finitely presented torsion-by-cyclic groups. Electronic Research Archive (2001).