6 GEORGIOS K. ALEXOPOULOS
1.4. The geometry of Lie groups of polynomial volume growth. We
present below the main families of Lie groups of polynomial volume growth and
we give some results on their geometry and the behavior at infinity of the centered
sub-Laplacians on these groups.
1. Simply connected nilpotent Lie groups. As we have already pointed out, all
connected nilpotent Lie groups have polynomial volume growth.
A special class of simply connected nilpotent Lie groups are the stratified ones
(cf. [FS]). These groups have a family of dilations. The dilation invariant sub-
Laplacians are the easiest ones to study.
Every simply connected nilpotent Lie group N behaves at infinity as a stratified
one NQ (cf. [NRS, VI]). Furthermore, any centered left invariant sub-Lapacian L
on N is converging at infinity to a dilation invariant sub-Laplacian LQ on NQ.
2. Non-simply connected nilpotent Lie groups. Let A" be a connected nilpotent
Lie group and let us assume that it is not simply connected. Let C be the maximal
torus of A". C is a compact central analytic subgroup of N and N/C is simply
connected (cf. [Ho p. 188, Va pp. 195-200]). Let us denote by ix the quotient map
7r : N — N/C. Then, the limit group at infinity AQ of A" is the same as the limit
group at infinity (N/C)o of N/C and the limit sub-Laplacian L0 of L is the limit
sub-Laplacian d7r(L)o of its image dTr(L) on N/C.
3. The semidirect product N X M of a simply connected nilpotent Lie group
N by a compact Lie group M. As one might expect, the group N \ M has the
same behavior at infinity as the group N. But the action of M on A' gives rise to
phenomena of homogenisation (cf. [Al, A2]). So the limit operator associated to
a left invariant sub-Laplacian L on N X M is called homogenised and denoted by
LH- It is a left invariant sub-Laplacian on N and is given by a formula similar to
the formula that one has in the classical homogenisation theory (cf. [BLP, JKO]).
This formula involves certain smooth functions ipi, which are called correctors and
which are defined on M.
4. Simply connected solvable Lie groups. Let S be such a group (of polynomial
volume growth) and let L be a centered left invariant sub-Laplacian on S. As all
connected Lie groups, S has a largest normal analytic nilpotent subgroup N S,
called nil-radical (cf. [Va]). We have that [S,S] C N. So, S/N is a simply
connected abelian Lie group and hence it can be identified with
We would like to view L as an operator on a nilpotent Lie group SN and thus
reduce its study to the nilpotent case.
The simplest case is when S splits into a semidirect product S = A'XR^, with
the action of
on N being semisimple. This is the case for example when S is
algebraic. Then, as SN we take the direct product SN = N xRk.
If S does not split into a semidirect product then the construction of SN is
more complicated. Intuitively, this is done as follows. We consider first a convenient
section of S/N which we identify again with Mfc. The action of this section on N
has a semisimple and a unipotent part. If we chose to ignore the semisimple part
and keep only the unipotent part then we obtain a nilpotent group denoted by SN.
We shall call SN nil-shadow of S. This is done in analogy with Auslander
and Green [AG], who used the term nil-shadow to describe the nilpotent part in
the semisimple splitting of S. The semisimple splitting gives an imbedding of a
connected solvable Lie group S into another solvable Lie group S' which is the
semidect product S' = Ns X A of a nilpotent Lie group Afe, the nil-shadow of 5,