We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.

中文翻译：

---

关于 p-ADIC 简单体积的注释

我们定义和研究单纯体积在任意半范数环上的推广，重点是p-adic 单纯卷。我们研究对素数的依赖性并根据以下公式建立同调界p-adic 单纯卷。作为主要示例，我们计算失重和p- 表面的 adic 单纯体积。这是基于在没有双曲线矫直的情况下计算曲面的经典单纯体积的另一种方法，并表明曲面满足 modp和p- 单纯体积的进阶近似。