A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a GBS group). Every GBS group is the fundamental group π1(𝔸) of a suitable labeled graph 𝔸. We prove that if 𝔸 and 𝔹 are labeled trees, then the groups π1(𝔸) and π1(𝔹) are universally equivalent iff π1(𝔸) and π1(𝔹) are embeddable into each other. An algorithm for verifying universal equivalence is pointed out. Moreover, we specify simple conditions for checking this criterion in the case where the centralizer dimension is equal to 3.

中文翻译：

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广义 Baumslag-Solitar 群的普遍等价

作用在树上的有限生成群使得所有顶点和边稳定器都是无限循环群，称为广义 Baumslag-Solitar 群（GBS 群）。每个 GBS 群都是一个合适的标记图 𝔸 的基本群 π1(𝔸)。我们证明如果 𝔸 和 𝔹 是标记树，那么群 π1(𝔸) 和 π1(𝔹) 是普遍等价的，如果 π1(𝔸) 和 π1(𝔹) 可以相互嵌入。提出了一种验证普遍等价的算法。此外，我们指定了在扶正器维数等于 3 的情况下检查该标准的简单条件。