Snowflake groups and conjugator length functions with non-integer exponents

TL;DR

This paper explores novel geometric phenomena in conjugacy problems of discrete groups, demonstrating that certain groups exhibit conjugator length functions and annular Dehn functions with non-integer exponents, leading to dense spectra of growth rates.

Contribution

The authors construct new groups with conjugator length functions and Dehn functions having non-integer exponents, revealing dense spectra of growth behaviors in geometric group theory.

Findings

Conjugator length functions in snowflake groups grow linearly with n.

Annular Dehn functions exhibit growth rates of n^{2α} with non-integer α.

Constructed groups have dense sets of exponents in their growth spectra.

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 $\text{Ann}(n) \simeq n^{2\alpha}$, 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 $[2,\infty)$.