Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups

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 $\mathrm{\Psi }:N\to M$. The displacement cocycle of $\mathrm{\Psi }$ and the right-invariant means on $N$ determine a compact convex set $\mathcal{A}(\mathrm{\Psi })\subseteq \mathrm{Hom}(V_N,V_M)$. If ${\mathcal{G}}_{\mathrm{p}\mathrm{t}}(\mathrm{\Psi })$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $\mathrm{\Psi }$ with varying basepoints, then $$\mathcal{A}(\mathrm{\Psi })=\mathrm{conv}{\mathcal{G}}_{\mathrm{p}\mathrm{t}}(\mathrm{\Psi }),\mathrm{Ext}\mathcal{A}(\mathrm{\Psi })\subseteq {\mathcal{G}}_{\mathrm{p}\mathrm{t}}(\mathrm{\Psi })\mathrm{.}$$ For every extreme point $L\in \mathcal{A}(\mathrm{\Psi })$, 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.