Uniform equicontinuity and groups of homeomorphisms

If G is a group of homeomorphisms of a uniform space (X, L) and the action is uniformly equicontinuous, then the topologies of pointwise tau p and uniform tau L convergences are among admissible group topologies. We investigate uniform properties of topological groups (G, tau(p)) and (G, tau(L)) of homeomorphisms of a uniform space X with uniformly equicontinuous action, uniform properties of X and connections between them. If X is a coset space of G with respect to a neutral subgroup and the maximal equiuniformity U on X is totally bounded, then the action is uniformly micro-transitive. Necessary and sufficient conditions when the group of homeomorphisms in the topology of pointwise convergence is kappa-narrow (in particular precompact) are given. Spectral representations of acting groups and phase spaces are presented. A sufficient condition for the Roelcke precompactness of a topological group is established. For the actions of the unitary group on the unit sphere in a Hilbert space and of the isometry group on the Urysohn sphere U1 in the topology of pointwise convergence the maximal equiuniformities are totally bounded. The maximal equivariant compactification beta U-G(1) is homeomorphic to the Hilbert cube. (C) 2021 Elsevier B.V. All rights reserved.