Conjugator Lengths and Isoperimetric functions
Abstract
We show that for any recognizing S-machine S with superadditive time function f, there exists a finitely presented group whose conjugator length function is quadratic and whose Dehn function grows like the square of f. This result is dual to that of a previous paper of the authors, which constructed a family of groups with cubic Dehn function that have various conjugator length functions. We thereby give the first known example of a finitely presented group whose conjugator length function is recursive, but whose Word and Conjugacy Problems are both undecidable. This answers an analogue of a question of Rips. Moreover, given a finitely presented group G with decidable Word Problem, we obtain a finitely presented group with decidable Conjugacy Problem whose Dehn function grows faster than that of G. Finally, combining this result with its dual, for a wide array of pairs of functions (f,g) we furnish an example of a finitely presented group with Dehn function equivalent to f and conjugator length function equivalent to g. This shows that the two invariants are very strongly independent and making significant progress on a question of Bridson, Riley, and Sale.
Related
BibTeX
@misc{gillis2026conjugator,
title = {{Conjugator Lengths and Isoperimetric functions}},
author = {Conan Gillis and Francis Wagner},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30879},
eprint = {2608.30879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30879},
}
RIS
TY - UNPB
TI - Conjugator Lengths and Isoperimetric functions
AU - Conan Gillis
AU - Francis Wagner
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30879
UR - https://arxiv.org/abs/2608.30879
AB - We show that for any recognizing S-machine S with superadditive time function f, there exists a finitely presented group whose conjugator length function is quadratic and whose Dehn function grows like the square of f. This result is dual to that of a previous paper of the authors, which constructed a family of groups with cubic Dehn function that have various conjugator length functions. We thereby give the first known example of a finitely presented group whose conjugator length function is recursive, but whose Word and Conjugacy Problems are both undecidable. This answers an analogue of a question of Rips. Moreover, given a finitely presented group G with decidable Word Problem, we obtain a finitely presented group with decidable Conjugacy Problem whose Dehn function grows faster than that of G. Finally, combining this result with its dual, for a wide array of pairs of functions (f,g) we furnish an example of a finitely presented group with Dehn function equivalent to f and conjugator length function equivalent to g. This shows that the two invariants are very strongly independent and making significant progress on a question of Bridson, Riley, and Sale.
ER -