Seminarium 30.10.2025
Piotr Niemiec
(Uniwersytet Jagielloński)
The classical theorems of Gao and Kechris (2002); Melleray (2008); and Solecki and Malicki (2009) assert that every Polish group (respectively: every compact Polish group; every locally compact Polish group) is isomorphic to the full isometry group of some separable complete (respectively: compact; proper) metric space. However, these theorems do not provide information about which metrizable space $X$ can be chosen to model a given Polish group $G$ as the isometry group of $X$.
During the talk, we shall present a complete characterization of topological groups that admit a model in the form of the full isometry group of some metric space (complete or not). For groups modeled by isometry groups of complete (respectively: compact; locally compact separable) metric spaces of bounded topological weight, we will indicate a specific (respectively: compact; connected locally compact) completely metrizable space which may serve as an underlying space for modeling such groups.
We will also present results describing which sets of transformations of a fixed compact (or, more generally, separable locally compact) metrizable space coincide with the full isometry group of that space with respect to some compatible metric.
If time permits, we will also discuss the most recent results about modeling groups equipped with bi-invariant metrics (up to isometric isomorphism) by full isometry groups endowed with the supremum metric.
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