A note on $p$-adic simplicial volumes

TL;DR

This paper introduces and explores $p$-adic simplicial volumes, generalizing classical concepts to seminormed rings, and provides explicit computations for surfaces, revealing new approximation properties and computational methods.

Contribution

It defines $p$-adic simplicial volumes, investigates their properties, and computes them for surfaces, offering new tools for calculating classical simplicial volume without hyperbolic straightening.

Findings

Computed $p$-adic simplicial volumes for surfaces.

Established homology bounds in terms of $p$-adic simplicial volumes.

Showed surfaces satisfy mod $p$ and $p$-adic approximation of simplicial volume.

Abstract

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.