Abstract:Invariant norms appear in most branches of mathematics. Examples include word norms (autonomous, entropy, fragmentation) and non-discrete norms (Hofer norm) in symplectic geometry. In group theory examples include commutator length and primitive length. After providing some history and motivation, I will focus on a group $G$ of volume-preserving diffeomorphisms of a compact smooth orientable manifold. In particular, I will discuss $L^p$-geometry of this group and which groups can be quasi-isometrically embedded in $G$. This talk is partially based on a joint work with M. Marcinkowski and E. Shelukhin.