Denote the Baumslag–Solitar family of groups as [Formula: see text]). When [Formula: see text] we study the Bass–Serre tree [Formula: see text] for [Formula: see text] as a geometric object. We suggest that the irregularity of [Formula: see text] is the principal obstruction for computing the growth series for the group. In the particular case [Formula: see text] we exhibit a set [Formula: see text] of normal form words having minimal length for [Formula: see text] and use it to derive various counting algorithms. The language [Formula: see text] is context-sensitive but not context-free. The tree [Formula: see text] has a self-similar structure and contains infinitely many cone types. All cones have the same asymptotic growth rate as [Formula: see text] itself. We derive bounds for this growth rate, the lower bound also being a bound on the growth rate of [Formula: see text].

中文翻译：

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Baumslag–Solitar 群的增长 II：Bass–Serre 树

将 Baumslag-Solitar 群族表示为 [公式：见正文]）。当 [Formula: see text] 我们将 Bass-Serre 树 [Formula: see text] 研究为 [Formula: see text] 作为几何对象。我们认为[公式：见正文]的不规则性是计算组增长序列的主要障碍。在特定情况下 [Formula: see text] 我们展示了一组 [Formula: see text] 具有最小长度的范式单词 [Formula: see text] 并使用它来推导各种计数算法。语言 [公式：见正文] 是上下文相关的，但不是上下文无关的。树[公式：见正文]具有自相似结构，包含无限多的锥体类型。所有锥体都具有与 [公式：见正文] 本身相同的渐近增长率。我们得出这个增长率的界限，