A Database of One-relator Groups

I put together a database of one-relator groups during my PhD. You can download a copy in CSV format below.

This database contains all $\mathrm{Aut}(F(\Sigma))$ representatives up to length 8, and all $\mathrm{Aut}(F(\Sigma))$ representatives where $|\Sigma| = 2$ 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.

Setup

We first fix an alphabet $\Sigma = \{a, b, c, ...\}$ which we will use for our generators. Capital letters $\Sigma^{-1} = \{A, B, C, ...\}$ will denote inverse generators and we will endow the set $\Sigma \cup \Sigma^{-1}$ 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 $W(\Sigma)$. Let $w, v \in W(\Sigma)$, then $w < v$ if $|w| < |v|$, or if $|w| = |v|$ and $w[{:}i] = v[{:}i]$ but $w[i] < v[i]$. This ordering descends to an ordering on freely reduced words in the free group $F(\Sigma)$. The automorphism group $\mathrm{Aut}(F(\Sigma))$ partitions $F(\Sigma)$ into orbits, each with a unique smallest element according to our order.

Let $\langle a, b, ... \mid r(a, b, ...)\rangle$ be a presentation of a one-relator group $G$. For each $\phi \in \mathrm{Aut}(F(\Sigma))$, we have that $\langle a, b, ... \mid \phi(r(a, b, ...))\rangle$ 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 $\mathrm{Aut}(F(\Sigma))$. It is well known (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

Relator: Minimal relator under the action of $\mathrm{Aut}(F(\Sigma))$ defining the group.
Name: The name of the group.
Number of Generators: The number of generators.
Torsion: False if the 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 $\langle a, b, ... \mid r(a, b, ...)\rangle$ 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 $\langle a, b, ... \mid r(a, b, ...)\rangle$. This property is preserved in the $\mathrm{Aut}(F_n)$ 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.

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.