I am putting together a database of one-relator groups, you can download a copy from here in csv format.
We first fix an alphabet S = {a, b, c, …} which we will use for our generators. Capital letters S^{-1} = {A, B, C, …} will denote inverse generators and we will endow the set S U S^{-1} with a total order given by
a < b < c < … < A < B < C < …
We may also define a total order on the set of words over our alphabet (with inverses), known as the shortlex order. We will denote this set by W(S). Let w, v be in W(S), then w < v if |w| < |v| or if |w| = |v| and w[:i] = v[:i] but w[i] < v[i] for some i. This ordering descends to an ordering on freely reduced words in the free group F(S). The automorphism group Aut(F(S)) partitions F(S) into orbits, each with a unique smallest element according to our order.
Let <a, b, … | r(a, b, …)> be a presentation of a one-relator group G. For each f in Aut(F(S)), we have that <a, b, … | f(r(a, b, …))> is also a presentation for G. Hence, given a one-relator presentation, we may obtain a minimal presentation by minimising the length of our relator under the action of Aut(F(S)). It is a well known result (Proposition 5.13 in [LS]) that a one-relator group is freely indecomposable if a minimal presentation has defining relation involving all of the generators. We only include one-relator groups which are freely indecomposable, hence the generators are precisely those that appear in its minimal relator.
The entries for the database are currently:
| Relator: | Minimal relator under the action of Aut(F(S)) defining the group. |
| Name: | The name of the group. |
| Number of Generators: | The number of generators. |
| Torsion: | False if group is torsion free, the root of the relator if it has torsion. If the group has torsion then its relator is a proper power and all torsion is conjugate into the subgroup generated by the root of the relator. One-relator groups with torsion are hyperbolic. |
| Abelianisation: | The abelianisation of the group. |
| Small Cancellation: | True if the presentation is a small cancellation presentation, False otherwise. Small cancellation groups are hyperbolic. |
| Centre: | A generating set for the centre of the group. The algorithm used for computing this is due to Baumslag and Taylor and may be found in [BT]. One-relator groups with non trivial centre are free-by-cyclic and automatic. |
| Geometric: | True if the relator is a geometric word, False if it is not a geometric word and nan if not known. Let <a, b, … | r(a, b, …)> be a one-relator group with say n generators. Then r(a, b, …) is a geometric word if there is a simple closed curve on the boundary of the handlebody of genus n which represents the word. By attaching a 2-handle along this curve we obtain a 3-manifold with boundary whose fundamental group is isomorphic to <a, b, … | r(a, b, …)>. This property is preserved in the Aut(F(S)) orbit of the given presentation. These entries were computed using John Berge’s software Heegaard. |
| Manifold, Knot, Link Exterior: | If the group appears as the fundamental group of a 3-manifold in the snappy census, then the value of this column is the identifier. |
| Further Information: | Any additional information such as other names and properties of the group. |
| References: | A list of references in which the group is mentioned. |
The database contains all Aut(F(S)) representatives up to length 8 and all Aut(F(S)) representatives where |S| = 2 up to length 14. If you have any suggestions for improving the database or would like to contribute new entries, feel free to drop me an email.
References:
[BT] G. Baumslag, T. Taylor. The Centre of Groups with One Defining Relator. Math. Annalen 175, 315,–319 (1968).
[LS] R.C. Lyndon, P. E. Schupp. Combinatorial Group Theory. Springer-Verlag. 2000.