Computational Explorations in Thompson’s Group F

Burillo, José; Cleary, Sean; Wiest, Bert

doi:10.1007/978-3-7643-8412-8_2

Cited by 13 publications

(24 citation statements)

References 19 publications

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“…In [7], Burillo, Cleary and Wiest used probabilistic methods to estimate the moments m n in Case 2 (T = A + A −1 + B + B −1 ) for n = 10, 20, . .…”

Section: In Particular This Extrapolation Methods Predicts Thatmentioning

confidence: 99%

“…, 11 can easily be computed. We will comment in more detail on the results of [7,14] at the end of Sec. 4.…”

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

confidence: 98%

“…[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%

“…Our paper is the first that considers the moments of T * 1 T 1 for T 1 := I + A+ B. In [7] Burillo, Cleary and Wiest use probabilistic methods to estimate the moments of m n (T * 2 T 2 ) = m 2n (T 2 ) for T 2 := A + A −1 + B + B −1 for n = 10, 20, 30, . .…”

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

confidence: 99%

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A computational approach to the Thompson group F

Haagerup¹,

Haagerup

Ramirez-Solano

2015

Int. J. Algebra Comput.

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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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“…In [7], Burillo, Cleary and Wiest used probabilistic methods to estimate the moments m n in Case 2 (T = A + A −1 + B + B −1 ) for n = 10, 20, . .…”

Section: In Particular This Extrapolation Methods Predicts Thatmentioning

confidence: 99%

“…, 11 can easily be computed. We will comment in more detail on the results of [7,14] at the end of Sec. 4.…”

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

confidence: 98%

“…[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%

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

confidence: 99%

See 2 more Smart Citations

A computational approach to the Thompson group F

Haagerup¹,

Haagerup

Ramirez-Solano

2015

Int. J. Algebra Comput.

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“…The group F is finitely presented. 10,21,22,8]). The commutator subgroup of F is an infinite simple group.…”

Section: Introductionmentioning

confidence: 99%

2-chains and square roots of Thompson’s group

Koberda¹,

Lodha²

2019

Ergod. Th. Dynam. Sys.

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We study two-generated subgroups x f, gy ă Homeo`pIq such that x f 2 , g 2 y is isomorphic to Thompson's group F, and such that the supports of f and g form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by C 2 diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by PL homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to M. Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of F as part of a general phenomenon among subgroups of Homeo`pIq.

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