We study topological groups of the form $L_0(\phi, G)$, where $\phi$ is a submeasure, $G$ is a topological group, and $L_0(\phi, G)$ carries the convergence-in-submeasure topology. We use a new concentration of measure theorem and a new classification of submeasures to investigate the existence of fixed points for continuous actions of $L_0(\phi,G)$ on compact spaces. Our theorems here unify and extend earlier results of Herer-Christensen, Glasner, and Pestov.

This is joint work with F. Martin Schneider.