Dehn functions of Lie groups
joint with Romain Tessera
Sample application to finitely generated groups: The Dehn function of a polycyclic group is either exponential or at most polynomial.
1. Basic facts
Dehn function makes sense for compactly presented locally compact groups. Compactly presented means that the groups admits a presentation of the form where both and are compact. E.g. connected Lie groups.
Theorem 1 (Bridson) If is a simply connected Lie group, its filling function (arising from a compact presentation) is equivalent to its Riemannian filling function.
Theorem 2 (Gromov) Connected Lie groups have at most exponential Dehn function.
Indeed, any connected Lie group is quasi-isometric to a simply connected solvable Lie group which admits a retraction by deformation with at most exponential distorsion.
2. The rank one obstruction
Gromov’s general upper bound is sharp.
Theorem 3 (Thurston) has exponential Dehn function.
is the semi-direct product of by acting by diagonal matrices of determinant 1.
Proof uses integration of an invariant 2-form. The optimal extension of the method yields
Theorem 4 Let be a triangulable group (i.e. a connected closed subgroup of the group of upper triangular real matrices). Assume that it admits two principal weights . Then Dehn function is exponential.
Note that every connected Lie group is quasi-isometric to a triangulable group. Note that the “rank one obstruction” in above theorem arises iff is onto for some . is the semi-direct product of by acting by diagonal matrices diag.
Definition 5 Let be a triangulable group. Its Lie algebra admits a canonical grading, called the Cartan grading , such that
– is a Cartan algebra (nilpotent and self-normalized),
– for all , the restriction of on has eigenvalues .
The sub-algebra generated by all , non zero, is called the exponential radical of . It generates the subgroup of exponentially distorted elements.
The weights of the abelianization of are called principal weights.
3. The homological obstruction
There is a second source of exponential behaviour, arising from the following remark: If there is a central extension with exponentially distorted, then has a exponential Dehn function.
Theorem 6 If is non zero, then Dehn function is exponential.
We call this the “homological obstruction”.
On the other hand, Gromov gave examples of polycyclic groups containing but having quadratic Dehn function.
Theorem 7 Let be triangulable. If neither rank one or homological obstructions arise, then the Dehn function is at most polynomial.
The corollary about polycyclic groups follows.
Assume no obstruction arises.
1. Reduction to (semi-direct product) where is abelian. Then has at most cubic Dehn function. If moreover Kill, then Dehn function is quadratic.
First case: If weights are contained in a half-space (say is tame), some element of contracts , so Dehn function is at most quadratic.
Otherwise, one constructs a group as an amalgam of tame subgroups. Then (Abels in -adic case) is a central extension of . If furthermore Kill, is isomorphic to .