Snowflake groups and conjugator length functions with non-integer exponents

Abstract

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 $\text{Ann}(n)\simeq n^{2\hat{I}\pm }$, where $\hat{I}\pm ={\log }_2(2p/q)$. Then, building on $B_{pq}$, we construct groups ${\tilde{B}}_{pq}^{+}$, for which $\text{CL}(n)\simeq n^{\hat{I}\pm +1}$. Thus the conjugator length spectrum and the spectrum of exponents of annular Dehn functions are both dense in the range $[2,\infty )$.

27 pages, 5 figures