2022/12/05 by Neretin, Yury A.
Mathematics · #18B10 #20E08 #20M20 #22A25 #54D35 #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2212.02607
openalex publication_date 2022/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider groups \mathbbI of isometries of ultrametric Urysohn spaces \mathbbU. Such spaces \mathbbU admit transparent realizations as boundaries of certain R-trees and the groups \mathbbI are groups of automorphisms of these R-trees. Denote by \mathbbI[X]⊂ \mathbbI stabilizers of finite subspaces X⊂ \mathbbU. Double cosets \mathbbI[X]⋅ g⋅ \mathbbI[Y], where g∈ \mathbbI, are enumerated by ultrametrics on union of spaces X∪ Y. We construct natural associative multiplications on double coset spaces \mathbbI[X]\backslash \mathbbI/\mathbbI[X] and, more generally, multiplications \mathbbI[X]\backslash \mathbbI/\mathbbI[Y] × \mathbbI[Y]\backslash \mathbbI/\mathbbI[Z]→ \mathbbI[X]\backslash \mathbbI/\mathbbI[Z]. These operations are a kind of canonical amalgamations of ultrametric spaces. On the other hand, this product can be interpreted in terms of partial isomorphisms of certain R-trees (in particular, we come to an inverse category). This allows us to classify all unitary representations of the groups \mathbbI and to prove that groups \mathbbI have type I. We also describe a universal semigroup compactification of \mathbbI whose image in any unitary representation of \mathbbI is compact.