Groups, Geometry, and Dynamics

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Volume 8, Issue 3, 2014, pp. 747–774
DOI: 10.4171/GGD/246

Published online: 2014-10-02

Relative amenability

Pierre-Emmanuel Caprace[1] and Nicolas Monod[2]

(1) Université Catholique de Louvain, Belgium
(2) Ecole Polytechnique Fédérale de Lausanne, Switzerland

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem.

We study the class $\mathscr{X}$ of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that $\mathscr{X}$ contains all familiar groups. Actually, no group is known to lie outside $\mathscr{X}$.

Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class $\mathscr{X}$.

Keywords: Amenability, subgroups, Chabauty topology, approximate identity

Caprace Pierre-Emmanuel, Monod Nicolas: Relative amenability. Groups Geom. Dyn. 8 (2014), 747-774. doi: 10.4171/GGD/246