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A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

Given a saturated fusion system F on a finite p-group S we define a ring A(F ) modeled on the Burnside ring A(G) of finite groups. We show that these rings have several properties in common. When F is the fusion system of G we describe the relationship between these rings.

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.