Geometry of growth: Approximation theorems for L^2 invariants

TL;DR

This paper develops approximation theorems for L^2 invariants, connecting infinite-dimensional topological invariants with finite-dimensional analogues, and introduces new invariants like torsion dimension to refine these relations.

Contribution

It generalizes Lück's theorem to broader contexts, introduces a new torsion dimension invariant, and studies growth processes in the setting of von Neumann categories.

Findings

Established approximation theorems linking infinite and finite dimensional invariants.

Introduced torsion dimension as a new invariant of torsion objects.

Showed torsion dimension vanishes under certain arithmetic conditions.

Abstract

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of L\"uck, dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite $p$-groups ($p$ is a prime), where we use the homology with coefficients in a finite field $\fp$. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion objects.