On spectral measures for certain unitary representations of R. Thompson's group F

Abstract Following a procedure due to Jones, using suitably normalized elements in a Temperley–Lieb–Jones (planar) algebra, we introduce a 3-parametric family of unitary representations of the Thompson’s group $F$ equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behavior at infinity of their matrix coefficients, thus showing that these representations do not contain any finite-type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of $F$. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of $F$ indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the 1st non-trivial index value for which the corresponding subgroup is isomorphic to the Brown–Thompson’s group $F_3$, we show that when the index is large enough, this subgroup is always trivial.

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Thompson’s Group F

Compact Textbooks in Mathematics - A Sampling of Remarkable Groups ◽

10.1007/978-3-030-01978-5_2 ◽

2018 ◽

pp. 37-66

Author(s):

Marianna C. Bonanome ◽

Margaret H. Dean ◽

Judith Putnam Dean

Keyword(s):

Thompson’S Group ◽

Thompson's Group F

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ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

International Journal of Algebra and Computation ◽

10.1142/s021819670400192x ◽

2004 ◽

Vol 14 (05n06) ◽

pp. 677-702 ◽

Cited By ~ 8

Author(s):

V. S. GUBA

Keyword(s):

Growth Rate ◽

Lower Bound ◽

Cayley Graph ◽

Upper Bound ◽

Growth Function ◽

Vertex Degree ◽

General Growth ◽

Thompson’S Group ◽

Average Vertex ◽

Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

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Random subgroups of Thompson’s group F

Groups Geometry and Dynamics ◽

10.4171/ggd/76 ◽

2010 ◽

pp. 91-126 ◽

Cited By ~ 5

Author(s):

Sean Cleary ◽

Murray Elder ◽

Andrew Rechnitzer ◽

Jennifer Taback

Keyword(s):

Thompson’S Group ◽

Thompson's Group F

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On Jones’ subgroup of R. Thompson’s group T

International Journal of Algebra and Computation ◽

10.1142/s0218196718500388 ◽

2018 ◽

Vol 28 (05) ◽

pp. 877-903

Author(s):

Jordan Nikkel ◽

Yunxiang Ren

Keyword(s):

Unitary Representation ◽

Unitary Representations ◽

Thompson Groups ◽

Diagram Groups ◽

Thompson’S Group ◽

Planar Algebra ◽

Thompson Group

Jones introduced unitary representations for the Thompson groups [Formula: see text] and [Formula: see text] from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups [Formula: see text] and [Formula: see text]. Golan and Sapir studied [Formula: see text] and identified it as a copy of the Thompson group [Formula: see text]. In this paper, we completely describe [Formula: see text] and show that [Formula: see text] coincides with its commensurator in [Formula: see text], implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to [Formula: see text], showing that [Formula: see text] and [Formula: see text] are not isomorphic, but as annular diagram groups they have very similar presentations.

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