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 $\text{Ann}(n) \simeq n^{2α}$, where $α= \log_2(2p/q)$. Then, building on $B_{pq}$, we construct groups $\tilde{B}_{pq}^+$, for which $\text{CL}(n)\simeq n^{α+1}$. Thus the conjugator length spectrum and the spectrum of exponents of annular Dehn functions are both dense in the range $[2,\infty)$.

💡 Research Summary

This paper investigates the complexity of the conjugacy problem in finitely presented groups through the lens of two quantitative functions: the conjugator length function (CL(n)) and the annular Dehn function (Ann(n)). The main achievement is proving that the spectra of exponents for these functions (considering growth of the form n^α) are both dense in the interval