Many transformation groups on manifolds are simple, but their universal coverings are not. In the present paper, we study the concept of relatively simple group, that is, a group with the maximum proper normal subgroup. We show that many examples of universal coverings of transformation groups are relatively simple, including the universal covering $\widetilde{\mathrm{Ham}}(M,\omega )$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega )$. Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We generalize his construction to relatively simple groups, and study their large scale geometric structure. In particular, Tsuboi’s metric space of $\widetilde{\mathrm{Ham}}(M,\omega )$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,\omega )$.

In this paper, we show that the pointwise finite-dimensional two-parameter persistence module ${\mathbb{HF}}_{\ast }^{(\bullet ,\bullet ]}$, defined in terms of interlevel filtered Floer homology, is rectangle-decomposable. This allows for the definition of a barcode consisting only of rectangles in ${\mathbb{R}}^2$ associated with the bipersistence module. We observe that this rectangle barcode contains information about Usher’s boundary depth and spectral invariants developed by Oh, Schwarz, and Viterbo. Moreover, we introduce a novel invariant extracted from the rectangle barcode, which proves instrumental in detecting periodic solutions of Hamiltonian systems. Additionally, we establish relevant stability results, particularly concerning the bottleneck distance and Hofer’s distance.