We introduce a notion of Ricci curvature for Cayley graphs that can be thought of as “medium-scale” because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. We argue that it gives the foundation for a definition of Ricci curvature well adapted to geometric group theory, beginning by observing that the sign can easily be characterized in terms of conjugation in the group. With this conjugation curvature κ, abelian groups are identically flat, and in the other direction we show that κ ≡ 0 implies the group is virtually abelian. Beyond that, κ captures known curvature phenomena in right-angled Artin groups (including free groups) and nilpotent groups, and has a strong relationship to other group-theoretic notions like growth rate and dead ends. We study dependence on generators and behavior under embeddings, and close with directions for further development and study.

中文翻译：

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凯莱图的共轭曲率

我们为凯莱图引入了 Ricci 曲率的概念，它可以被认为是“中等尺度”，因为它既不是无穷小也不是渐近的，而是基于选择的有限半径参数。我们认为它为很好地适应几何群论的 Ricci 曲率定义奠定了基础，首先观察到符号可以很容易地根据群中的共轭来表征。有了这个共轭曲率 κ，阿贝尔群同样是平坦的，并且在另一个方向上我们证明了κ ≡ 0意味着该群实际上是阿贝尔群。除此之外，κ捕获直角 Artin 群（包括自由群）和幂零群中的已知曲率现象，并且与其他群论概念（如增长率和死胡同）有很强的关系。我们研究了对生成器的依赖和嵌入下的行为，并与进一步发展和研究的方向密切相关。