Unitary Representations of Oligomorphic Groups
By Todor Tsankov. 2011
84 citations recorded by the source corpus.
We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S∞, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand–Raikov theorem holds for topological subgroups of S∞: for all such groups, continuous irreducible representations separate points in the group.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/397329b8804e8df92304aa2bb7b900e3c1c55a92
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
- I. Yaacov
- Terry Lyons
- Ilya Chevyrev
- Julien Melleray
- L. N. V. Th'e
- Todor Tsankov
- Rostislav I. Grigorchuk
In Mathematical Physics
- Ilya Chevyrev
- I. Yaacov
- Terry Lyons
- L. N. V. Th'e
- Todor Tsankov
- Julien Melleray
- Peter K. Friz
- Antoine Lejay
Citations in the corpus
- Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups (2013, result): “ur as automorphism groups of ℵ0-categorical structures: they are exactly the Roelcke precompact Polish groups. This generalizes a similar result about automorphism groups of classical structures from [T1]. Definition 1.1. A topological group G is called Roelcke precompact if for every neighbourhood U”
- Polish groups with metrizable universal minimal flows (2014, background): “mains open in general. Zucker’s result also implies that Question 1.1 is equivalent to the question whether every oligomorphic permutation group has ametrizable universal minimal flow. It was shown in [T] that oligomorphic permutation groups are (more or less) the Roelcke precompact subgroups of S∞. ”
- Characteristic functions of measures on geometric rough paths (2013, background): “ier transform always exists for locally compact groups by the Gelfand-Raikov theorem [15], unitary representations in the general case are known to exhibit a much more varied behavior (see e.g. [24], [31]). Furthermore, classical harmonic analysis on locally compact maximally almost periodic groups ”