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+ε, for one fixed ε>0. We use a convex invariant of a fixed quasi-isometry Ψ:N→ M. The displacement cocycle of Ψ and the right-invariant means on N determine a compact convex set 𝒜(Ψ)⊆Hom(VN,VM). If 𝒢pt(Ψ) denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of Ψ with varying basepoints, then 𝒜(Ψ)=conv𝒢pt(Ψ), Ext𝒜(Ψ)⊆𝒢pt(Ψ). For every extreme point L∈𝒜(Ψ), the dual map L* 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.
Related
- Quasi-isometries between graphs of totally disconnected locally compact groups
- Additive quasi-isometries and cacti
- Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces
- Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries
- A nilpotent quotient algorithm for certain infinitely presented groups and its applications
BibTeX
@misc{mukherjee2026cohomology,
title = {{Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups}},
author = {Mayukh Mukherjee and Soumyadeb Samanta and Soumyadip Thandar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13055},
eprint = {2610.13055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13055},
}
RIS
TY - UNPB
TI - Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups
AU - Mayukh Mukherjee
AU - Soumyadeb Samanta
AU - Soumyadip Thandar
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13055
UR - https://arxiv.org/abs/2610.13055
AB - 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+ε, for one fixed ε>0. We use a convex invariant of a fixed quasi-isometry Ψ:N→ M. The displacement cocycle of Ψ and the right-invariant means on N determine a compact convex set 𝒜(Ψ)⊆Hom(V_N,V_M). If 𝒢_pt(Ψ) denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of Ψ with varying basepoints, then 𝒜(Ψ)=conv𝒢_pt(Ψ), Ext𝒜(Ψ)⊆𝒢_pt(Ψ). For every extreme point L∈𝒜(Ψ), the dual map L^* 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.
ER -