AI 中文总结
本文研究循环群的图积的$\ell^2$-Betti数与$\Sigma$-不变量，将其与右角Artin群关联，并在有限阶顶点结构特定情形下给出显式公式，同时计算自同构群的相关不变量。
AI 中文摘要
我们研究循环群的图积的$\ell^2$-Betti数。我们将这些不变量与右角Artin群的$\ell^2$-Betti数联系起来，并在由定义图中有限阶顶点的结构决定的若干情形下获得显式公式。特别地，我们考虑这些顶点构成团或具有相同链接的情形，并在定义图恰好有两个有限阶顶点时推导出第一个$\ell^2$-Betti数的公式。我们还计算了当定义图至多有两个有限阶顶点时，循环群的图积的BNSR-不变量及其自同构群的第一个$\ell^2$-Betti数。
英文摘要
We study the $\ell^2$-Betti numbers of graph products of cyclic groups. We relate these invariants to the $\ell^2$-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first $\ell^2$-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first $\ell^2$-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.
Comments22 pages
