Counting elements and geodesics in Thompson's group F

Communicated by M. SapirLet F denote the Thompson group with standard generators A = x 0 , B = x 1 . It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent towhere in both cases the norm of an element in the group ring CF is computed in B( 2 (F )) via the regular representation of F . By extensive numerical computations, we obtain precise lower bounds for the norms in (i) and (ii), as well as good estimates of the spectral distributions of (I + A + B) * (I + A + B) and of A + A −1 + B + B −1 with respect to the tracial state τ on the group von Neumann Algebra L(F ). Our computational results suggest, thatIt is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F .

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“…[7,3,12,14,13]). Our paper is the first that considers the moments of T * 1 T 1 for T 1 := I + A+ B.…”

Section: Moreover In Both Cases the Upper Bound Is Attained If And Omentioning

confidence: 99%

“…It is known (cf. [12,17]) that the size of the spheres S n of radius n in the word metric of F , grows roughly with a factor φ 2 = 2.618 . .…”

Section: Preliminariesmentioning

confidence: 99%

A computational approach to the Thompson group F

Haagerup¹,

Haagerup

Ramirez-Solano

2015

Int. J. Algebra Comput.

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“…Although by Theorem 5.6.42 and Exercise 5.6.46, the R. Thompsongroup F admits a Church-Rosser presentation with a rational language of canonical words, we cannot conclude that the growth series of F (say, with respect to the generating set {x 0 , x 1 }) is represented by a rational function as in Section 1.8.9 because the canonical words are not geodesic. But Elder, Fusy and Rechnitzer [96] formulated the following The word "typical" which should be intuitively clear, can be precisely defined (see [168], for example). The exact values of the spherical growth function s(n) of F (with respect to {x 0 , x 1 }) are known for n ≤ 1500 [96].…”

Section: Diagram Groupsmentioning

confidence: 99%

“…But Elder, Fusy and Rechnitzer [96] formulated the following The word "typical" which should be intuitively clear, can be precisely defined (see [168], for example). The exact values of the spherical growth function s(n) of F (with respect to {x 0 , x 1 }) are known for n ≤ 1500 [96]. This data implies that if the growth series of F with respect to the generating set {x 0 , x 1 } is represented by a rational function P Q, then the degree of the polynomial Q must be at least 749.…”

Section: Diagram Groupsmentioning

confidence: 99%

Combinatorial Algebra: Syntax and Semantics

Sapir¹,

Guba²,

Volkov³

2014

Springer Monographs in Mathematics

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“…. Leaf and caret labels used in [6]. Leaves are labeled "N" or "I" and carets are labeled "n" or "i."…”

Section: A 2-counter Language Of Normal Forms For Elements Of Fmentioning

confidence: 99%

Tree-based language complexity of Thompson's group F

Taback

Younes²

2015

Groups Complexity Cryptology

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The definition of graph automatic groups by Kharlampovich, Khoussainov and Miasnikov and its extension to C-graph automatic by Murray Elder and the first author raise the question of whether Thompson's group F is graph automatic. We define a language of normal forms based on the combinatorial "caret types" which arise when elements of F are considered as pairs of finite rooted binary trees, which we show to be accepted by a finite state machine with 2 counters, and forms the basis of a 3-counter graph automatic structure for the group.2010 Mathematics Subject Classification. 20F65, 68Q45.

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Counting elements and geodesics in Thompson's group F

Cited by 12 publications

References 24 publications

A computational approach to the Thompson group F

A computational approach to the Thompson group F

Combinatorial Algebra: Syntax and Semantics

Tree-based language complexity of Thompson's group F

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