---
abstract: 'We prove an analogue of the Approximation Theorem of $L^2$-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.'
address:
- |
    Department of Mathematics\
    Virginia Tech\
    Blacksburg, VA 24061-0123,\
    USA
- |
    Westfälische Wilhelms-Universität Münster\
    Mathematisches Institut\
    Einsteinstr. 62, D-48149 Münster, Germany
- |
    Westfälische Wilhelms-Universität Münster\
    Mathematisches Institut\
    Einsteinstr. 62, D-48149 Münster, Germany
author:
- Peter Linnell
- Wolfgang Lück
- Roman Sauer
date: 'March 1, 2010'
title: 'The limit of $\mathbf{\IF_p}$-Betti numbers of a tower of finite covers with amenable fundamental groups'
---

Introduction
============

A *residual chain* of a group $G$ is a sequence $G=G_0\supset G_1\supset G_2\supset \cdots$ of normal subgroups of finite index such that $\bigcap_{i\ge 0}G_i=\{e\}$. The $n$-th $L^2$-Betti number of any finite free $G$-CW complex $X$ is the limit of the $n$-th Betti numbers of $G_i\backslash X$ normalized by the index $[G:G_i]$ for $i\to\infty$ [@Lueck(1994c)]. If we instead consider Betti numbers $b_n(G_i\backslash X;k)$ with respect to a field of characteristic $p>0$, the questions whether the limit exists, what it is, and whether it is independent of 