  We investigate a family of groups acting on a regular tree, defined by
prescribing the local action almost everywhere. We study lattices in these
groups and give examples of compactly generated simple groups of finite
asymptotic dimension (actually one) not containing lattices. We also obtain
examples of simple groups with simple lattices, and we prove the existence of
(infinitely many) finitely generated simple groups of asymptotic dimension one.
We also prove various properties of these groups, including the existence of a
proper action on a CAT(0) cube complex.
