Conjugator length in nilpotent groups

RESUMO

For every rational number $\alpha\geq 2$, we construct a 2-step nilpotent group with conjugator length function $\operatorname{CL}(n)\simeq n^\alpha$. We deduce that the nilpotent conjugator length spectrum is dense in $\{0\}\cup[2,\infty)$, 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 $m \geq 1$, the cocompact lattices in $\mathsf H(\mathbb R)^m$ realize exactly the growth types $n^2,\ldots,n^{m+1}$. Finally, for every real cubic algebraic number $\theta$, we construct a 2-step nilpotent group such that $n^3 \preceq \operatorname{CL}(n) \preceq n^{3+\varepsilon}$ for every $\varepsilon > 0$, with $n^3 \prec \operatorname{CL}(n)$ if and only if at least one real conjugate of $\theta$ 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.