Mathematics – Dynamical Systems
Scientific paper
2009-09-05
Mathematics
Dynamical Systems
17 pages; corrected typos
Scientific paper
The following two results are shown. 1) Let $G$ be the $k$-rational points of a simple algebraic group over a local field $k$ and let $H$ be a lattice in $G.$ Then the regular representation of $G$ on $L^2(G/H)$ has a spectral gap (that is, there are no almost invariant unit vectors in the subspace of functions in $L^2(G/H)$ with zero mean). 2) There exist locally compact simple groups $G$ and lattices $H$ for which $L^2(G/H)$ has no spectral gap. This answers in the negative a question asked by Margulis. In fact, $G$ can be taken to be the group of orientation preserving automorphisms of a $k$-regular tree for $k>2.$
Bekka Bachir
Lubotzky Alexander
No associations
LandOfFree
Lattices with and lattices without spectral gap does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Lattices with and lattices without spectral gap, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lattices with and lattices without spectral gap will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-161167