Quasiisometries of nilpotent groups
Joint work (in progress) with David Kyed. The paper is not fully written, so be careful.
1. Nilpotent groups
Let be finitely generated torsion free nilpotent. Mal’cev’s theorem states that there exists a unique connected Lie group where sits as a discrete cocompact subgroup.
This is not hard for the free step nilpotent group, the general case follows. See Baumslag’s notes.
Fact. Any two cocompact lattices in a Lie group are quasiisometric. Therefore nilpotent group having the same Mal’cev completion are quasiisometric.
Question. Is the converse true ?
2. Earlier results
In 1989, Pansu showed that quasiisometric nilpotent groups have the same graded Lie algebras. Given a Lie algebra , let denote the descending central series. Then
with the induced Lie bracket.
In 2002, Shalom showed that quasiisometric nilpotent groups have the same usual Betti numbers. Apart from degree one, this does not follow from the previous result.
2.1. Sketch of Shalom’s argument
Two groups and are quasiisometric iff they are uniformly measure equivalent.
Definition 1 and are uniformly measure equivalent if they have commuting measure preserving actions on some measure space , and both action are co-finite. Furthermore, pick a fundamental domain with resulting cocycle ; one requires that for fixed , is bounded.
In this case, cohomology of the module can be transferred from to .
Next, Shalom uses Property (see below). This implies that
3. Cohomology and higher order cohomology
Definition 2 A continuous -module is relatively injective if for every exact sequence
of continuous -modules, which admits a continuous linear section (when this is the case, we speak of a strengthened morphism), also admits a -equivariant continuous linear section.
Proposition 3 For every -module , the module of continuous equivariant maps is relatively injective.
A strengthened resolution of follows:
Definition 4 Let be a -module. Pick a relatively injective resolution. The continuous cohomology of is the cohomology of the subcomplex of -invariant vectors of the resolution.
Note that we do not take closures.
3.2. Higher order invariants
Definition 5 For and , denote by
For instance, invariant vectors
Definition 6 For and , denote by
The functor is left-exact. As for all left-exact functors, one can define its polynomial cohomology.
Definition 7 Let be a -module. Pick an arbitrary relative injective resolution
Its polynomial cohomology in degree , , is the cohomology of the resulting complex of order invariants .
As an exercise, let us compute . iff for all and ,
I.e. is a homomorphism plus a constant.
Elements of are known as bi-characters
polynomials of degree . Whence the name polynomial cohomology.
3.4. Polynomial maps
More generally, Lazard (’50s), Leibman (2002) call elements of polynomial maps on . Note that polynomials can be multiplied together.
Now I describe polynomials in the nilpotent Lie group case. Fix a Mal’cev basis, i.e. a linear basis adapted to the descending central filtration. Denote . Then the map
is a diffeomorphism. Each coordinate becomes a function on , this is a polynomial map of degree .
Theorem 9 nilpotent Lie group. Then is generated as an algebra by functions .
Example. Heisenberg group. Then
4. Main result
Theorem 10 (Kyed-Petersen) Let , be nilpotent Lie groups. Assume they are uniformly measure equivalent. Then for all and ,
Let be the group multiplication. One can show that it induces which is a complete invariant of . Our strategy is to express in terms of polynomial cohomology. In this way, we hope to be able to prove that quasiisometric nilpotent Lie groups must be isomorphic.
Recall that uniform measure equivalence means there exists a measure space with a action, isomorphisms to and respectively, with and compact.
Theorem 11 (Reciprocity theorem, inspired by Monod-Shalom)
Then Property leads to
which relates to the higher cohomology of . Then recursion.