Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

2025/03/05 by Danilenko, Alexandre I., Dudko, Artem
#22D10 #37A40 #Dynamical Systems (math.DS) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2503.03894

Abstract

Let G be a countable branch group of automorphisms of a spherically homogeneous rooted tree. Under some assumption on finitarity of G, we construct, for each sequence ω∈\0,1\^\Bbb N, an irreducible unitary representation κ_ω of G. Every two representations κ_ω and κ_ω' are weakly equivalent. They are unitarily equivalent if and only if ω and ω' are tail equivalent. Each κ_ω appears as the Koopman representation associated with some ergodic G-quasiinvariant measure (of infinite product type) on the boundary of the tree.

Related

• On irreducibility of Koopman representations of Higman-Thompson groups
• Groupoid actions and Koopman representations
• Criteria of irreducibility of the Koopman representations for the group \rm GL₀(2∞,\mathbb R)
• Characterising actions on trees yielding non-trivial quasimorphisms
• On diagonal actions of branch groups and the corresponding characters