Conjugation Curvature in Solvable Baumslag-Solitar Groups

TL;DR

This paper investigates conjugation curvature in Baumslag-Solitar groups, providing explicit geodesic words, a length formula, and demonstrating the existence of elements with positive, negative, and zero curvature at positive density.

Contribution

It introduces a formula for word length in $BS(1,n)$ and shows the prevalence of different conjugation curvature types among elements.

Findings

Explicit geodesic words for elements in $BS(1,n)$

A formula for word length in the group

Existence of elements with positive, negative, and zero curvature at positive density

Abstract

For an element in $BS(1,n) = \langle t,a | tat^{-1} = a^n \rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w \geq 0$ and $v \in \mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set $\{t,a\}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.