An ergodic theorem for the quasi-regular representation of the free group

2016/01/04 by Boyer, Adrien, Lobos, Antoine Pinochet
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1601.00668

Abstract

In \citeBAMU, an ergodic theorem à la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \citeBOYER to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \citeROBLI, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group \mathbbF_r of rank r even if \mathbbF_r is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.

Related

• Equidistribution, ergodicity and irreducibility in CAT(-1) spaces
• Equidistribution, ergodicity and irreducibility associated with Gibbs\n measures
• Non-removable term ergodic action semigroups/groups
• Ergodic boundary representations
• Geometric covering arguments and ergodic theorems for free groups