## Metric differentiation, monotonicity and maps to L^1

Cheeger, Jeff
Kleiner, Bruce
##### Description
We give a new approach to the infinitesimal structure of Lipschitz maps into L^1. As a first application, we give an alternative proof of the main theorem from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1. The proof uses the metric differentiation theorem of Pauls and the cut metric decomposition to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group.
Comment: Added a missing condition to Definition 5.3. Also made a number of minor corrections, and added a reference to a paper by Lee-Raghavendra
##### Keywords
Mathematics - Metric Geometry, Mathematics - Differential Geometry, Mathematics - Functional Analysis, Mathematics - Group Theory