# Disjoint superheavy subsets and fragmentation norms

## TL;DR

This paper establishes a lower bound for the fragmentation norm in Hamiltonian diffeomorphism groups and constructs a bi-Lipschitz embedding, addressing a question about fragmentation norms on surfaces.

## Contribution

It introduces a lower bound for the fragmentation norm and constructs a bi-Lipschitz embedding into the Hamiltonian diffeomorphism group, answering a specific open question.

## Key findings

- Established a lower bound for the fragmentation norm.
- Constructed a bi-Lipschitz embedding of imensional space into imensional Hamiltonian diffeomorphism group.
- Provided an answer to Brandenbursky's question on fragmentation norms for surfaces.

## Abstract

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$ with respect to the fragmentation norm on the group $\mathrm{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$. As an application, we provide an answer to Brandenbursky's question on fragmentation norms on $\mathrm{Ham}(\Sigma_g)$, where $\Sigma_g$ is a closed Riemannian surface of genus $g\geq 2$