  This work presents a general paradigm to construct finite combinatorial
objects, with extermal substructures. This is done by observing that there
exist infinite combinatorial structures with desired extermal substructures,
coming from an action of a subgroup of the automorphism group. The crux of our
idea, that we call the closed orbits method is a systematic way to construct a
finite quotient of the infinite structure, containing a simple shadow of the
infinite substructure, which maintains its extermal combinatorial property. In
particular, the shadow of a thin infinite substructure in the cover remains
small in the quotient.
  We give a couple of applications of our paradigm. The first application is to
the well-known question of vertex expansion of number theoretic Ramanujan
graphs. It has been a common belief that such graphs are lossless expanders. We
show that some number theoretic Ramanujan graphs are not even unique neighbor
expanders, which is a weaker notion. This holds, in particular, for the
Morgenstern Ramanujan graphs.
  The second application is in the field of graph quantum ergodicity, where we
produce number theoretic Ramanujan graphs with an eigenfunction of small
support that corresponds to the zero eigenvalue. This again contradicts common
expectations.
  The closed orbits method is based on an established idea from dynamics and
number theory, of studying closed orbits of subgroups. The novelty of this work
is in exploiting this idea to combinatorial questions.
