Calculus on nilpotent Lie groups (Q2908715)

The author considers the group \(UTU(n, \mathbb R)\) of upper triangular unipotent \({n\times} n\)-matrices (equipped with its natural stratification) and shows the following rigidity properties of Pansu differentiable maps.NEWLINENEWLINETheorem 1. Let \(n\geq 4\). Suppose that the map \(f:UTU(n, \mathbb R)\to UTU(n, \mathbb R)\) is Pansu differentiable at a point \(p\) with a Pansu differentiable local inverse at \(f(p)\). Then the Pansu derivative \(df (p )\) is either given via conjugation by a diagonal matrix (this is denoted by \(df( p)\in \operatorname{Ad} (MA)\)) or in \(\phi \circ \operatorname{Ad} (MA)\), where \(\phi\) is the reflection ant-diagonal (that is, \(\phi(g)_{ij}=g_{n+1-j,n+1-i}\)).NEWLINENEWLINETheorem 2: Let \(n\geq 3\). Suppose that the continuous map \(f:UTU(n, \mathbb R)\to UTU(n, \mathbb R)\) is Pansu differentiable almost everywhere, and that \(df\) is in \(\operatorname{Ad} (MA)\) almost everywhere. Then \(f\) is an affine map, that is, the composition of a group automorphism (that commutes with dilations) and a translation.NEWLINENEWLINEThe proofs are obtained by direct calculations for \(n=3\) and \(4\) in Theorems 1 and 2, respectively, followed by induction.NEWLINENEWLINEIn the introduction and in the conclusion, the author reviews various related results and motivations, especially from Mostow's rigidity theorem.NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].