When the holonomy group is trivial, [GAMMA] can be considered to be a lattice in G and the corresponding manifold [GAMMA]\G is a nilmanifold. When G is abelian, i.e.
It is known that every map on a nilmanifold is weakly Jiang, due to the result of Anosov () or Fadell and Husseini ().
Therefore, every infranilmanifold of the form [GAMMA]\G is finitely covered by a nilmanifold [conjunction]\G, such that every continuous map f : [GAMMA]\G [right arrow] [GAMMA]\G can be lifted to a map [bar.f] : [conjunction]\G [right arrow] [conjunction]\G.
This actually means that if we choose good representatives in the Reidemeister classes of [[[[beta].sub.r]].sub.r] and [[[[beta].sub.s]].sub.s], p'(Fix([[beta].sub.n][[~.g].sup.n])) will reduce to both p'(Fix([[beta].sub.r][[~.g].sup.r])) and p'(Fix([[beta].sub.s][[~.g].sup.s])) on our nilmanifold [LAMBDA]\G.
[MATHEMATICAL EXPRESSION OMITTED] then M is diffeomorphic to a noncompact nilmanifold
As shown in [1, 2, 7, 8, 3] (see also Theorems 2.1 and 2.2) there are simple formulas for the numbers N([f.sup.m]), N[P.sub.m](f) and N[[PHI].sub.m](f) for fixed m on tori and nilmanifolds. These formulas involve the linearization F of f (see  and section 2.1), which any self map of a torus or nilmanifold possesses.
Of course we are thinking of A as a self map of nilmanifold or torus.
In the first part of this subsection we remind the reader of the concept of linearization, of a self map of a torus or nilmanifold, details can be found in [3,10].
A striking example is the role of dynamics on nilmanifolds
in the recent proof of Hardy-Littlewood estimates for the number of solutions to systems of linear equations of finite complexity in the prime numbers.
McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds
and solvmanifolds II, Topology Appl., 75 (1997), 81-?92.
 Edward Keppelmann, Periodics points on nilmanifolds
and solvmanifolds, Pacific Journal of Mathematics, vol.164 (1) (1994), 105-128.
In , flat and nilmanifolds
whose fundamental groups possess property [R.sub.[infinity]] were constructed.