Conjugator length in nilpotent groups

Abstract

For every rational number $\alpha \ge 2$, we construct a 2-step nilpotent group with conjugator length function $\mathrm{CL}(n)\simeq n^{\alpha }$. 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 $m\ge 1$, the cocompact lattices in $\mathsf{H}(\mathbb{R}{)}^m$ realize exactly the growth types $n^2,\dots ,n^{m+1}$. Finally, for every real cubic algebraic number $\theta$, we construct a 2-step nilpotent group such that $n^3⪯\mathrm{CL}(n)⪯n^{3+\epsilon }$ for every $\epsilon >0$, with $n^3\prec \mathrm{CL}(n)$ 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.

21 pages. Comments are welcome!