Local spectral gap in simple Lie groups and applications

2015/03/22 by Boutonnet, Rémi, Ioana, Adrian, Golsefidy, Alireza Salehi
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1503.06473

Abstract

We introduce a novel notion of \it local spectral gap for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action Γ\curvearrowright G, whenever Γ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group G. This extends to the non-compact setting recent works of Bourgain and Gamburd \citeBG06,BG10, and Benoist and de Saxcé \citeBdS14. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on G. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique Γ-invariant finitely additive measure defined on all bounded measurable subsets of G.

Related

• Spectral Independence
• Local spectral gap in the group of Euclidean isometries
• Spectral rigidity of group actions on homogeneous spaces
• Strongly ergodic actions have local spectral gap
• A spectral gap theorem in simple Lie groups