Conjugator length in nilpotent groups
Abstract
For every rational number α≥ 2, we construct a 2-step nilpotent group with conjugator length function CL(n)≃ nα. We deduce that the nilpotent conjugator length spectrum is dense in {0}∪[2,∞), even after restricting to groups of nilpotency class at most $2$. Moreover, we show that, despite being a commensurability invariant of nilpotent groups, conjugator length is not a quasi-isometry invariant: for every m ≥ 1, the cocompact lattices in H(ℝ)m realize exactly the growth types n2,…,nm+1. Finally, for every real cubic algebraic number θ, we construct a 2-step nilpotent group such that n3 ⪯ CL(n) ⪯ n3+ε for every ε > 0, with n3 ≺ CL(n) if and only if at least one real conjugate of θ has unbounded partial quotients. Consequently, determining the conjugator length function for arbitrary 2-step nilpotent groups is at least as hard as settling the bounded-partial-quotient problem for real cubic algebraic numbers with two nonreal conjugates.
Related
BibTeX
@misc{dere2026conjugator,
title = {{Conjugator length in nilpotent groups}},
author = {Jonas Deré and Ken Vandermeersch},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14190},
eprint = {2609.14190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14190},
}
RIS
TY - UNPB
TI - Conjugator length in nilpotent groups
AU - Jonas Deré
AU - Ken Vandermeersch
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14190
UR - https://arxiv.org/abs/2609.14190
AB - For every rational number α≥ 2, we construct a 2-step nilpotent group with conjugator length function CL(n)≃ n^α. We deduce that the nilpotent conjugator length spectrum is dense in {0}∪[2,∞), even after restricting to groups of nilpotency class at most $2$. Moreover, we show that, despite being a commensurability invariant of nilpotent groups, conjugator length is not a quasi-isometry invariant: for every m ≥ 1, the cocompact lattices in H(ℝ)^m realize exactly the growth types n^2,…,n^m+1. Finally, for every real cubic algebraic number θ, we construct a 2-step nilpotent group such that n^3 ⪯ CL(n) ⪯ n^3+ε for every ε > 0, with n^3 ≺ CL(n) if and only if at least one real conjugate of θ has unbounded partial quotients. Consequently, determining the conjugator length function for arbitrary 2-step nilpotent groups is at least as hard as settling the bounded-partial-quotient problem for real cubic algebraic numbers with two nonreal conjugates.
ER -