$L^p$ measure equivalence of nilpotent groups

TL;DR

This paper classifies nilpotent groups up to $L^p$ measure equivalence for all $p\, extless=1$, revealing new rigidity phenomena and constructing explicit orbit equivalences using advanced geometric and cohomological techniques.

Contribution

It introduces new methods to classify nilpotent groups under $L^p$ measure equivalence, including explicit orbit constructions and rigidity results for $p>1$.

Findings

Existence of $L^p$ orbit equivalences between simply connected nilpotent Lie groups for some $p>0$

Characterization of when groups have isomorphic asymptotic cones in terms of $L^p$ ME

Examples of nilpotent groups with isomorphic Carnot graded groups that are not $L^p$ ME for some finite $p$

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.