$L^2$-Betti numbers and computability of reals

TL;DR

This paper investigates the computability of real numbers derived from $L^2$-Betti numbers and torsion in groups, focusing on how their computability relates to the Turing degree of the word problem.

Contribution

It introduces a framework connecting $L^2$-invariants of groups with computability theory, exploring how these invariants' degrees depend on the group's word problem.

Findings

Establishes links between $L^2$-invariants and Turing degrees.

Shows certain $L^2$-Betti numbers are non-computable.

Provides insights into the complexity of invariants in geometric group theory.

Abstract

We study the computability degree of real numbers arising as $L^2$-Betti numbers or $L^2$-torsion of groups, parametrised over the Turing degree of the word problem.