For an element in $BS(1,n)=〈t,a\mid ta{t}^{-1}=a^n〉$ written in the normal form $t^{-u}{a}^v{t}^w$ with $u,w\ge 0$ 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.

中文翻译：

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可解 Baumslag-Solitar 群中的共轭曲率

对于一个元素$乙\mathrm{小号}(\mathrm{1个},n)=〈吨,A\forall 吨A{吨}^{-\mathrm{1个}}=A^n〉$写成正常形式${吨}^{-你}{A}^v{吨}^w$和$你,w\ge 0$和$v\in \mathbb{Z}$，我们展示了一个代表元素的测地线词，并给出了它相对于生成集的词长的公式$\{吨,A\}$. 使用这个字长公式，我们证明存在正密度正密度元素集，即正共轭曲率、负共轭曲率和零共轭曲率，正如 Bar Natan、Duchin 和 Kropholler 所定义的那样。