Geodesic
On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees
Yury A. Neretin1
1University of Vienna, Austria
Groups, geometry, and dynamics, Tome 15 (2021) no. 3, pp. 801-824

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DOI

Consider an infinite homogeneous tree Tn​ of valence n+1, its group Aut(Tn​) of automorphisms, and the group Hier(Tn​) of its spheromorphisms (hierarchomorphisms), i.e., the group of homeomorphisms of the boundary of Tn​ that locally coincide with transformations defined by automorphisms. We show that the subgroup Aut(Tn​) is spherical in Hier(Tn​), i.e., any irreducible unitary representation of Hier(Tn​) contains at most one Aut(Tn​)-fixed vector. We present a combinatorial description of the space of double cosets of Hier(Tn​) with respect to Aut(Tn​) and construct a “new” family of spherical representations of Hier(Tn​). We also show that the Thompson group Th has PSL(2,Z)-spherical unitary representations.
DOI : 10.4171/ggd/612
Classification : 22-XX, 20-XX, 43-XX
Mots-clés : Spheromorphism, hierarchomorphism, spherical representation, spherical subgroup, infinite symmetric group, Bruhat–Tits tree

Yury A. Neretin  1

1 University of Vienna, Austria 
Yury A. Neretin. On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees. Groups, geometry, and dynamics, Tome 15 (2021) no. 3, pp. 801-824. doi: 10.4171/ggd/612
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     title = {On spherical unitary representations of groups of spheromorphisms of {Bruhat{\textendash}Tits} trees},
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     pages = {801--824},
     year = {2021},
     volume = {15},
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     doi = {10.4171/ggd/612},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/612/}
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