# Cohomological induction and uniform measure equivalence

## TL;DR

This paper develops a cohomological induction framework linking measure equivalence of groups to their cohomology, offering new tools for classifying nilpotent Lie groups up to quasi-isometry.

## Contribution

It introduces a general cohomological induction isomorphism based on uniform measure equivalence, unifying and extending previous results in the field.

## Key findings

- Graded cohomology rings of quasi-isometric nilpotent Lie groups are isomorphic.
- Unified results of Shalom and Sauer on cohomology and measure equivalence.
- Provides new insights into the quasi-isometry classification of low-dimensional nilpotent Lie groups.

## Abstract

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.