# A Database of One-relator Groups

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 $\Sigma = \{a, b, c, ...\}$ 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 $Aut(F(\Sigma))$ partitions $F(\Sigma)$ into orbits, each with a unique smallest element according to our order.

Let $\langle a, b, ... \vert r(a, b, ...)\rangle$ be a presentation of a one-relator group $G$. For each $\phi\in Aut(F(\Sigma))$, we have that $\langle a, b, ... \vert \phi(r(a, b, ...))\rangle$ 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 $Aut(F(\Sigma))$. 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 $Aut(F(\Sigma))$ 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 $\langle a, b, ...\vert r(a, b, ...)\rangle$ 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 $\langle a, b, ...\vert r(a, b, ...)\rangle$. This property is preserved in the $Aut(F_n)$ 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.

The database contains all $Aut(F(\Sigma))$ representatives up to length 8 and all $Aut(F(\Sigma))$ 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.

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.