$PSL(2,\mathbb{Z})$ as a non-distorted subgroup of Thompson's group T

Fossas, Ariadna

doi:10.1512/iumj.2011.60.4477

Cited by 8 publications

(5 citation statements)

References 6 publications

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“…May be the most strange is its relation with the Minkowski function ? (x) (apparently observed by Sergiescu, see [3]). ⊠ For any pair of two such half-planes there is a unique element of PSL 2 (Z) sending one half-plane to another.…”

Section: Introductionmentioning

confidence: 71%

“…Denote by Aut(T) the group of all automorphisms of T. We define the topology on Aut(T) assuming that all point-wise stabilizers K(J) of finite subtrees J ⊂ T are open subgroups in Aut(T). We get a Polish group 3 .…”

Section: Introductionmentioning

confidence: 99%

“…Remark. The Thompson group Th was defined by Richard Thompson in 1966 as a counterexample, later it became clear that it is a very interesting discrete group with unusual properties, see, e. g., [4], [6], [3]. May be the most strange is its relation with the Minkowski function ?…”

Section: Introductionmentioning

confidence: 99%

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On the group of spheromorphisms of a homogeneous non-locally finite tree

Neretin¹

2020

Izv. Math.

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Consider a tree T, all whose vertices have countable valence; its boundary is the Baire space B ≃ N N ; continued fractions expansions identify the set of irrational numbers R \ Q with B. Removing k edges from T we get a forest consisting of copies of T. A spheromorphism (or hierarchomorphism) of T is an isomorphisms of two such subforests considered as a transformation of T or of B. Denote the group of all spheromorphisms by Hier(T). We a show that the correspondence R \ Q ≃ B sends the Thompson group realized by piecewise PSL2(Z)-transformations to a subgroup of Hier(T). We construct some unitary representations of the group Hier(T), show that the group of automorphisms Aut(T) is spherical in Hier(T), and describe the train (enveloping category) of Hier(T).

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Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

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On the group of spheromorphisms of a homogeneous non-locally finite tree

Neretin¹

2020

Izv. Math.

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“…Topologies of this kind There is a well-known discrete group Th consisting of spheromorphisms 3 defined by 1965 R. Thompson in 1965. Initially it was proposed as a counterexample, and it really has strange properties but also it is an interesting positive object (see, e.g., [16], [6], [20], [50], [25], [4], [30], [11]).…”

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confidence: 99%

On spherical unitary representations of groups of spheromorphisms of Bruhat--Tits trees

Neretin

2019

Preprint

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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.

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“…The pictures are in hyperbolic geometry for clarity. Furthermore, Thurston noticed that Thompson's group T can be seen as the group of homeomorphisms of the real projective line which are piecewise P SL(2, Z)[23,14].…”

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confidence: 99%