Spectral theory of approximate lattices in nilpotent Lie groups

2018/11/15 by Michael Björklund, Björklund, Michael, Tobias Hartnick +1 · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Spectral Theory (math.SP) #math.DS #math.GR #math.SP

paper · pdf · doi:10.48550/arxiv.1811.06563

53 pages, Comments welcome!

arxiv created 2018/11/15 · arxiv updated 2018/11/19

Abstract

We show that an approximate lattice in a nilpotent Lie group admits a relatively dense subset of central (1-ε)-Bragg peaks for every ε> 0. For the Heisenberg group we deduce that the union of horizontal and vertical (1-ε)-Bragg peaks is relatively dense in the unitary dual. More generally we study uniform approximate lattices in extensions of lcsc groups. We obtain necesary and sufficient conditions for the existence of a continuous horizontal factor of the associated hull-dynamical system, and study the spectral theory of the hull-dynamical system relative to this horizontal factor.

Cited by

• Periodicity of point processes in abelian groups without lattices
• Siegel-Radon transforms of transverse dynamical systems

Related

• The structure of approximate groups
• The definitions of approximate lattices
• The structure of approximate lattices in linear groups
• Coherent systems over approximate lattices in amenable groups
• Sampling theorems for the Heisenberg groups