Conjugator Length in the Baumslag-Gersten Group

TL;DR

This paper investigates the conjugator length function in the Baumslag-Gersten group, revealing it grows faster than any fixed-height exponential tower, and compares it to iterated Baumslag-Solitar groups.

Contribution

It establishes bounds on the conjugator length function for the Baumslag-Gersten group and relates it to iterated exponential functions, providing new insights into its growth rate.

Findings

Conjugator length function grows faster than any fixed-height exponential tower.

Bounds are established showing exponential tower growth in the Baumslag-Gersten group.

Conjugator length in iterated Baumslag-Solitar groups is equivalent to iterated exponential functions.

Abstract

We show that the conjugator length function of the Baumslag-Gersten group is bounded above and below by a tower of exponentials of logarithmic height -- in particular it grows faster than any tower of exponentials of fixed height. We conjecture that no one-relator group has a larger conjugator length function than the Baumslag-Gersten group. Along the way, we also show that the conjugator length function of the $m$-th iterated Baumslag-Solitar groups is equivalent to the $m$-times iterated exponential function.