Séminaire de Géométrie, Groupes et Dynamique

Miklós Abert: "Co-spectral radius, equivalence relations and the growth of unimodular random rooted trees"

435 (UMPA)



We define the co-spectral radius of inclusions of discrete,
probability measure-preserving equivalence relations, as the sampling
exponent of a generating random walk on the ambient relation. This
generalizes the spectral radius for random walks on quotient Schreier
graphs with respect to subgroups. The almost sure existence of the
sampling exponent is already new for i.i.d. percolation clusters on
countable groups. For the proof, we develop a general method called
2-3-method that is based on the mass-transport principle. As a
byproduct, we show that the growth of a unimodular random rooted tree
of bounded degree always exists, assuming its upper growth passes a
critical threshold. This complements Timar's work who showed the
possible nonexistence of growth below this threshold. We also show
that the walk growth exists for an arbitrary unimodular random rooted
graph of bounded degree. We also investigate how the co-spectral
radius behaves for Property (T) and hyperfinite relations. This is
joint work with Mikolaj Fraczyk and Ben Hayes.