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Proper isometric actions of Thompson’s groups on Hilbert space. (English) Zbl 1113.22005

Int. Math. Res. Not. 2003, No. 45, 2409-2414 (2003); corrigendum ibid. 2023, No. 3, 2698-2700 (2023).
R. J. Thompson in unpublished notes of about 1965 introduced finitely presented discrete groups designated \(\mathcal{F}\), \(\mathcal{T}\) and \(\mathcal{V}\) which he used later to construct unsolvable word problems. The elements of \(\mathcal{V}\) may be represented as certain right continuous bijections on the unit interval which maps dyadic rational numbers to dyadic rational numbers. The group \(\mathcal{F}\) is the subgroup of those elements which are homeomorphisms and \(\mathcal{T}\) consists of those which induce homeomorphisms of the circle. These bijections can also be considered as piecewise linear mappings from domain \([0,1]\) to range \([0,1]\) and these can be represented (non-uniquely) by tree diagrams, see J.-P. Serre [Trees (Springer-Verlag, Berlin etc.) (1980; Zbl 0548.20018)]. A tree of standard dyadic intervals has as vertices the dyadic intervals and the edges are relevant pairs of dyadic intervals. To a standard dyadic partition with root the vertex \([0,1]\) there corresponds a tree with root \([0,1]\). The (unique) tree obtained by removing all unnecessary carets (i.e., leaves of the tree like a \(\wedge\)) is called reduced. J. W. Cannon, W. J. Floyd and W. R. Parry [Enseign. Math. 42, 215–256 (1996; Zbl 0880.20027)] proved that there is one and only one reduced tree diagram for any \(f \in \mathcal{F}\). The author extends this result to \(\mathcal{V}\). He then finds a condition, on representative triplets, made up of two standard dyadic partitions and a bijection between sets of intervals in the respective partitions, which ensures correspondence of a triple to a unique element of \(\mathcal{V}\).
An affine action mapping \(G\) to isometries of a Hilbert space \(H\) is a combination of an isometric representation \(\rho\) of \(g \in G\) and a representation \(\pi\) of \(G\) as translations on \(H\), the latter necessarily being associated to a 1-cocycle for \(\rho\). The action is called proper if for each \(r \in \mathbb R_{+}\) only a finite number of \(g\) can satisfy \(\|\pi(g)\| \leq r\). The author defines a discrete group to have property (T), after D. A. Kazhdan [Funct. Anal. Appl. 1, 63–65 (1967; Zbl 0168.27602)], if every affine isometric action has a global fixed point (equivalently, if there is a non-trivial invariant vector then there are no other almost-invariant vectors, as in J.-P. Serre [Astérisque 46 (1977; Zbl 0369.20013)] and P. de la Harpe and A. Valette [Astérisque 175 (1989; Zbl 0759.22001)]). We set the notation of this property as Kazhdan’s seeing that there are already too many T’s around.
It is an open question whether the Thompson groups are amenable, so the author concentrates on an easier question. A discrete group \(G\) is called a-T-amenable if there is a proper affine action on a Hilbert space. The term a-T-moyennable, for more general groups, was used by M. Gromov [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. London Mathematical Society Lecture Note Series 182. Cambridge University Press, Cambridge (1993; Zbl 0841.20039)] in the sense of essentially not having Kazhdan’s property, though Gromov’s property was initially used by R. Godement in the 50’s. It was already known that \(\mathcal{T}\) and \(\mathcal{F}\) do not have the Kazhdan property and that \(\mathcal{F}\) is a-T-amenable. In this article the author proves that \(\mathcal{V}\) (and so also \(\mathcal{T}\)), is a-T-amenable. Having constructed a suitable Hilbert space and cocycle and proving the uniqueness of a \(v\) corresponding to a triplet, the rest of the proof consists of showing, for any \(v\in \) \(\mathcal{V}\), that \(\pi\) is proper and \(\pi(v)\) is in the Hilbert space.
A few diagrams would have been helpful for those who do not have trees embedded in their brains.

MSC:

22D05 General properties and structure of locally compact groups
19K35 Kasparov theory (\(KK\)-theory)
05E25 Group actions on posets, etc. (MSC2000)
20G05 Representation theory for linear algebraic groups
05C05 Trees
43A07 Means on groups, semigroups, etc.; amenable groups
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