The expected degree of minimal spanning forests

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.

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Thom, A. The expected degree of minimal spanning forests. Combinatorica 36, 591–600 (2016). https://doi.org/10.1007/s00493-014-3160-x

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Mathematics Subject Classification (2000)

  • 60B99
  • 05C07
  • 05C80
  • 37A20