## 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. 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 ${G}$.