One-relator group census

A one-relator group is a group of the form \(F_r/\langle\langle w\rangle\rangle\) where \(F_r\) is the free group of rank \(r\) and \(w\) is word in the free group. If \(\alpha\in\mathrm{Aut}(F_r)\) then \(F_r/\langle\langle w\rangle\rangle\) and \(F_r/\langle\langle \alpha(w)^{\pm 1}\rangle\rangle\) are isomorphic, so the \(\mathrm{Aut}(F_r)\)–orbit of the cyclic subgroup \(\langle w\rangle\) determines the group. This is still redundant at the level of isomorphism type of one-relator group, but we don't know a convenient way to enumerate isomorphism types.

Let's define the length of such an automorphic class of cyclic subgroup to be the minimal length of a generator of a representative cyclic subgroup. We enumerate the automorphic orbits by length, for rank \(2\leq r\leq 4\) and length \(2\leq L\leq 16\) by choosing the shortlex minimal element among all cyclic generators of cyclic subgroups in the class. If the free group is, eg, \(F_4=\langle a,b,c,d\rangle\) then our base lexicographic order is \(D\lt C\lt B\lt A\lt a\lt b\lt c\lt d\), where case change denotes inversion.

This is based on the paper:
Cashen and Hoffmann, Short, highly imprimitive words yield hyperbolic one-relator groups, Experimental Mathematics (in press).

The code used to produce this data can be found on github.

Organization

For each \(r\in\{2,3,4\}\) we enumerate only words that are not contained in a proper free factor of \(F_r\). For each rank there is a single .csv file containing all representatives of length \(2\leq L\leq 13\), with some additional data. For \(14\leq L\leq 16\) the words are separated into .txt files by rank, length, and additionally by imprimitivity rank (irank). The imprimitivity rank of a word is the minimal rank of a subgroup of the free group containing the word as an imprimitive element. Some group information is determined by the imprimitivity rank: The one-relator group has torsion if and only if it has irank=1. For all data presented here, if the irank is not 2 then the one-relator group is hyperbolic. For irank=2 both hyperbolic and nonhyperbolic groups occur.

Stats

Total number of representatives by length and rank
length	\(F_1\)	\(F_2\)	\(F_3\)	\(F_4\)
1	1	0	0	0
2	1	0	0	0
3	1	0	0	0
4	1	2	0	0
5	1	3	0	0
6	1	8	1	0
7	1	12	5	0
8	1	34	18	2
9	1	71	98	5
10	1	217	522	35
11	1	515	3,124	315
12	1	1,423	16,866	7,106
13	1	3,834	96,086	93,460
14	1	11,816	582,844	1,124,764
15	1	33,321	3,481,458	11,679,597
16	1	95,440	19,514,686	109,264,221

Reps by rank and irank

@ length 14
irank	\(F_1\)	\(F_2\)	\(F_3\)	\(F_4\)
1	1	12	5	0
2	0	11,804	364	6
3	0	0	582,475	321
4	0	0	0	1,124,437

@ length 15
irank	\(F_1\)	\(F_2\)	\(F_3\)	\(F_4\)
1	1	3	0	0
2	0	33,318	258	7
3	0	0	3,481,200	1055
4	0	0	0	11,678,535

@ length 16
irank	\(F_1\)	\(F_2\)	\(F_3\)	\(F_4\)
1	1	34	18	2
2	0	95,406	2,765	111
3	0	0	19,511,903	11,023
4	0	0	0	109,253,085

Usage

The data is not human readable. This python module, orgcensus.py, will decode it.
Example:


$ python
>>> import orgcensus
>>> words=orgcensus.generator_from_text_file('Rank4Len16Irank1.txt',outputstyle='str')
>>> {w for w in words} 
set(['DDCCBBAADDCCBBAA', 'DCBAbadcDCBAbadc'])
>>> df=orgcensus.read_csv('Rank2Len2_13.csv',keystyle='str')
>>> df['length']=df.apply(lambda row: len(row.name),axis=1)
>>> df[df.length==4]
      irank hyperbolic hyperbolicparabolic hyperbolicreason  length
BBAA      2      False                None                c       4
BAba      2      False                None                i       4

The 'hyperbolicreason' codes are:
f - free
t - torsion
c - cyclically pinched
s - small cancellation
i - conditions from a paper of Ivanov and Schupp
b - condition from a paper of Blufstein and Minian
w - verified with GAP package walrus
k - verified with program kbmag

Data files

One-relator group census

Organization

Stats

Usage

Data files

Rank 2

Rank 3

Rank 4