Notes of Yves Cornulier’s lecture

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 {< S | R>} where both {S} and {R} are compact. E.g. connected Lie groups.

Theorem 1 (Bridson) If {G} 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) {Sol} has exponential Dehn function.

{Sol} is the semi-direct product of {\mathbb{R}^2} by {\mathbb{R}}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 {G} 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 {a<0<b}. 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 {G \rightarrow Sol_\lambda} is onto for some {\lambda}. {Sol_\lambda} is the semi-direct product of {\mathbb{R}^2} by {\mathbb{R}} acting by diagonal matrices diag{(e^t,e^{-\lambda t})}.

Definition 5 Let {G} be a triangulable group. Its Lie algebra admits a canonical grading, called the Cartan grading {\mathfrak{g}=\sum_a \mathfrak{g}_a}, such that

{\mathfrak{g}_0} is a Cartan algebra (nilpotent and self-normalized),

– for all {w \in \mathfrak{g}_0}, the restriction of {ad_w} on {\mathfrak{g}_a} has eigenvalues {e^{a(w)}}.

The sub-algebra {\mathfrak{g}^\infty} generated by all {\mathfrak{g}_a}, {a} non zero, is called the exponential radical of {\mathfrak{g}}. It generates the subgroup of exponentially distorted elements.

The weights of the abelianization of {\mathfrak{g}^\infty} 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 {1 \rightarrow \mathbb{R}\rightarrow \tilde{G} \rightarrow G \rightarrow 1} with {\mathbb{R}} exponentially distorted, then {G} has a exponential Dehn function.

Theorem 6 If {H_2(\mathfrak{g}^\infty)_0} 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 {Sol(\mathbb{Z})} but having quadratic Dehn function.

Theorem 7 Let {G} be triangulable. If neither rank one or homological obstructions arise, then the Dehn function is at most polynomial.

The corollary about polycyclic groups follows.

4. Proof

Assume no obstruction arises.

1. Reduction to {G=G^\infty \times D} (semi-direct product) where {D} is abelian. Then {G} has at most cubic Dehn function. If moreover Kill{(\mathfrak{g}^\infty)_0 = 0}, then Dehn function is quadratic.

First case: If weights are contained in a half-space (say {G} is tame), some element of {D} contracts {G^\infty}, so Dehn function is at most quadratic.

Otherwise, one constructs a group {\tilde{G}} as an amalgam of tame subgroups. Then (Abels in {p}-adic case) {\tilde{G}} is a central extension of {G}. If furthermore Kill{(\mathfrak{g}^\infty)_0 = 0}, {\tilde{G}} is isomorphic to {G}.


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