Conjugation curvature in solvable Baumslag–Solitar groups

For an element in [Formula: see text] written in the normal form [Formula: see text] with [Formula: see text] and [Formula: see text], we exhibit a geodesic word representing the element and give a formula for its word length with respect to the generating set [Formula: see text]. 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.