Snowflake groups and conjugator length functions with non-integer exponents

We exhibit novel geometric phenomena in the study of conjugacy problems for discrete groups. We prove that the snowflake groups $B_{pq}$, indexed by pairs of positive integers $p>q$, have conjugator length functions $\text{CL}(n)\simeq n$ and annular Dehn functions $\text{Ann}(n)\simeq n^{2\alpha }$, where $\alpha ={\log }_2(2p\mathrm{/}q)$. Then, building on $B_{pq}$, we construct groups ${\tilde{B}}_{pq}^{+}$, for which $\text{CL}(n)\simeq n^{\alpha +1}$. Thus the conjugator length spectrum and the spectrum of exponents of annular Dehn functions are both dense in the range $[2,\mathrm{\infty })$.