The expected degree of minimal spanning forests

TL;DR

This paper establishes a lower bound on the expected degree of the free minimal spanning forest in vertex transitive graphs using spectral radius, and explores implications for group theory and measurable group actions.

Contribution

It provides a new spectral radius-based lower bound for the expected degree of minimal spanning forests and applies this to group theory, answering open questions and simplifying existing proofs.

Findings

Lower bound on expected degree in terms of spectral radius

Answer to Lyons-Peres-Schramm question

Non-torsion unitarizable groups have fixed price one

Abstract

We give a lower bound on the expected degree of the free minimal spanning forest of a vertex transitive graph in terms of its spectral radius. This result answers a question of Lyons-Peres-Schramm and simplifies the Gaboriau-Lyons proof of the measurable-group-theoretic solution to von Neumann's problem. In the second part we study a relative version of the free minimal spanning forest. As a consequence of this study we can show that non-torsion unitarizable groups have fixed price one.