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 $\alpha = \log_2(2p/q)$. 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 $[2,\infty)$.