Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups

Abstract

We construct two word metrics on the same seven-dimensional torsion-free nilpotent lattice whose asymptotic cones are isometric, but for which every quasi-isometry has multiplicative constant at least $1+\epsilon$, for one fixed $\epsilon >0$. We use a convex invariant of a fixed quasi-isometry $\Psi :N\to M$. The displacement cocycle of $\Psi$ and the right-invariant means on $N$ determine a compact convex set $\mathcal{A}(\Psi )\subseteq \mathrm{Hom}(V_N,V_M)$. If ${\mathcal{G}}_{\mathrm{pt}}(\Psi )$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $\Psi$ with varying basepoints, then $$\mathcal{A}(\Psi )=\mathrm{conv}{\mathcal{G}}_{\mathrm{pt}}(\Psi ),\mathrm{Ext}\mathcal{A}(\Psi )\subseteq {\mathcal{G}}_{\mathrm{pt}}(\Psi ).$$ For every extreme point $L\in \mathcal{A}(\Psi )$, the dual map $L^{\ast }$ is also the degree-one part of an isomorphism of the full real cohomology rings. This cohomological compatibility, together with the realization of first-layer Pansu differentials by invariant means, yields the multiplicative constant gap stated above.

arXiv comment34 pages. This is split off from the authors' previous submission arXiv:2512.06753. Comments are most welcome!