$L^p$ measure equivalence of nilpotent groups
Thiebout Delabie, Claudio Llosa Isenrich, Romain Tessera
TL;DR
This work advances the measured and orbit-equivalence theory for groups of polynomial growth by connecting large-scale geometry (via Carnot gradings and Pansu limits) with ergodic- and cohomology-based rigidity. It proves a strong converse to Austin's $L^1$-ME rigidity: isomorphic Carnot-graded groups imply $L^p$ orbit equivalence for some $p>1$, and yields a full $L^1$ ME classification for compactly generated groups with polynomial growth. The authors develop constructive tools—explicit Følner tiling–based orbit couplings and entropy-controlled integrability criteria—alongside new obstructions from distorted central extensions to show limits of rigidity for $p>1$. They also extend these results to locally compact groups and lattices, providing quantitative bounds on the allowable $p$ For nilpotent pairs and highlighting central-extension obstructions that delimit $L^p$ ME in families like $G_{m,3}$ vs. $L_m\times\mathbb{R}^2$. Overall, the paper deepens the bridge between geometric group theory, measured group theory, and Lie theoretic invariants, with explicit constructions and clear directions for further tightening bounds on $p$ in $L^p$ ME/OE classifications.
Abstract
We classify compactly generated locally compact groups of polynomial growth up to $L^p$ measure equivalence (ME) for all $p\leq 1$. To achieve this, we combine rigidity results (previously proved for discrete groups by Bowen and Austin) with new constructions of explicit orbit equivalences between simply connected nilpotent Lie groups. In particular, we prove that for every pair of simply connected nilpotent Lie groups there is an $L^p$ orbit equivalence for some $p>0$, where we can choose $p>1$ if and only if the groups have isomorphic asymptotic cones. We also prove analogous results for lattices in simply connected nilpotent Lie groups. This yields a strong converse of Austin's Theorem that two nilpotent groups which are $L^1$ ME have isomorphic Carnot graded groups. We also address the much harder problem of extending this classification to $L^p$ ME for $p>1$: we obtain the first rigidity results, providing examples of nilpotent groups with isomorphic Carnot graded groups (hence $L^1$ OE) which are not $L^p$ ME for some finite (explicit) $p$. For this we introduce a new technique, which consists of combining induction of cohomology and scaling limits via the use of a theorem of Cantrell. Finally, in the appendix, we extend theorems of Bowen, Austin and Cantrell on $L^1$ ME to locally compact groups.