### Schauder's fixed point and amenability of a group

#### Abstract

A criterion for existence of a fixed point for an affine action

of a given group on a compact convex space is presented. From this we derive that

a discrete countable group is amenable if and only if there exists an invariant

probability measure for any action of the group on a Hilbert cube. Amenable properties of the

group of all isometries of the Urysohn universal homogeneous metric space

are also discussed.

of a given group on a compact convex space is presented. From this we derive that

a discrete countable group is amenable if and only if there exists an invariant

probability measure for any action of the group on a Hilbert cube. Amenable properties of the

group of all isometries of the Urysohn universal homogeneous metric space

are also discussed.

#### Keywords

Amenable groups; fixed points; invariant measures

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