A note on $p$-adic simplicial volumes
Steffen Kionke, Clara Loeh · University of Regensburg Publication Server (University of Regensburg) · 2020
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 gives a 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.