Nonunitarizable Representations and Random Forests

Abstract

Correspondence to be sent to: nicolas.monod@epfl.ch We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L 2 -Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroups.

Asymptotics of Cheeger constants and unitarisability of groups

“…Since unitarisability passes to subgroups, Dixmier's question thus concerns non-amenable groups without free subgroups. The fact that such groups can indeed be non-unitarisable has been confirmed more recently [12,24,29].…”

The expected degree of minimal spanning forests

“…Now, the desired result is an immediate consequence of (2) and [3,Theorem 1.3] from the work of Epstein-Monod.…”

“…We now change the perspective slightly and consider more general invariant random spanning forests. They are no longer bound to be sub-forests of G. Thus, we study more generally probability measures on {0, 1} Γ×Γ , invariant under the diagonal right Γ-action on Γ × Γ, see [3] for more details. The expected degree deg(σ) of an invariant random spanning forest σ defined in a similar way as before.…”

“…Our approach relies on results of Epstein-Monod [3], whose pioneering work related invariant random forests to this problem. Let us explain this in a bit more detail.…”

The Dixmier problem, lamplighters and Burnside groups

“…Recently, a criterion was discovered [4] that lead to examples without free subgroups (see [4,9]). We shall improve a strategy proposed in [7] in order to apply ergodic methods to the problem.…”