A Group from Beyond Infinity

The order topology on the first uncountable ordinal yields a fascinating topological space, well known for featuring "pathological" phenomena (such as sequential compactness without compactness, or a Stone–Čech compactification reduced to Alexandrov's one-point compactification). In this talk, we introduce its homeomorphism group, which also comes with its own niceties. For instance, it is an amenable topological group which is strongly allergic to any idea of distance: all its continuous isometric actions have bounded orbits and all its continuous morphisms to metrisable groups are trivial.