# Dynamical correspondences of $L^2$-Betti numbers

## TL;DR

This paper explores dynamical analogues of $L^2$-Betti numbers for modules over group rings, revealing their role in measuring the failure of addition formulas for dynamical invariants in sofic groups.

## Contribution

It introduces dynamical analogues of $L^2$-Betti numbers and demonstrates their significance in understanding the failure of addition formulas for dynamical invariants.

## Key findings

- $L^2$-Betti numbers measure failure of addition formula in dynamical invariants
- Establishes a connection between $L^2$-Betti numbers and dynamical properties of modules
- Provides new tools for analyzing modules over group rings in the context of sofic groups

## Abstract

We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the $L^2$-Betti numbers exactly measure the failure of addition formula for dynamical invariants.