orgcensus — One-relator group census

orgcensus — One-relator group census Census of one-relator groupsRanks 2–4, relator lengths 2–16 Description: This dataset contains the results of a census of the groups F_r / <<w>> with 2 ≤ r ≤ 4 and 2 ≤ |w| ≤ 16 Entries correspond to Aut(F_r) classes of cyclic subgroups <w> that are not contained in a proper free factor of F_r.Each equivalence class of cyclic subgroup is canonically represented by a single word generating it, where the chosen representative word is the shortlex minimal word among all those that can be obtained from it by applying elements of Aut(F_r) or inverting.The free groups are taken to be generated by lower case letters, with the corresponding upper case letter denoting the inverse of the generator, eg, F_4 is generated by a, b, c, d. The base lexicographic order we use is D<C<B<A<a<b<c<d. The additional group-theoretic invariants are included: irank imprimitivity rankhyperbolic True / False / Nonehyperbolicparabolic True / False / Nonehyperbolicreason string describing reason for hyperbolicity (e.g. small-cancellation, torsion, etc.) This census was the data source for the paper: CH. Cashen and C. Hoffmann,"Short, highly imprimitive words yield hyperbolic one-relator groups",Experimental Mathematics 32 (4), 631–640 (2023).DOI: https://doi.org/10.1080/10586458.2021.1982079 Mathematical context and consequences As a consequence of the result: Marco Linton,"One-relator hierarchies",Duke Math. J. 174 (4), 747–802 (2025).https://doi.org/10.1215/00127094-2024-0040 it is now known that every one-relator group is hyperbolic except when it has imprimitivity rank = 2, in which case both hyperbolic and non-hyperbolic examples exist. Files in the archive The archive consists of three CSV files (compressed with gzip): orgcensus_rank2.csv.gz all rank 2 equivalence classesorgcensus_rank3.csv.gz all rank 3 equivalence classesorgcensus_rank4.csv.gz all rank 4 equivalence classes Each CSV uses: index canonical word (string)columns irank, hyperbolic, hyperbolicparabolic, hyperbolicreason Loading the data with pandasimport pandas as pd df2 = pd.read_csv("orgcensus_rank2.csv.gz", index_col=0)df3 = pd.read_csv("orgcensus_rank3.csv.gz", index_col=0)df4 = pd.read_csv("orgcensus_rank4.csv.gz", index_col=0) Each DataFrame has: index canonical word (string)columns irank, hyperbolic, hyperbolicparabolic, hyperbolicreason Citation If you use this dataset, please cite this page and the paper: CH. Cashen and C. Hoffmann,"Short, highly imprimitive words yield hyperbolic one-relator groups",Experimental Mathematics 32 (4), 631–640 (2023).DOI: https://doi.org/10.1080/10586458.2021.1982079 Keywords one-relator group, free group, hyperbolic group, group theory, geometric group theory, census, dataset License (CC-BY 4.0) This dataset is released under the Creative Commons Attribution 4.0 International License (CC-BY 4.0).You may share and adapt the material, including commercially, provided appropriate credit is given. Full license text: https://creativecommons.org/licenses/by/4.0/