Random Sampling of Trivial Words in Finitely Presented Groups

Elder, Murray; Rechnitzer, Andrew; Rensburg, E J Janse van

doi:10.1080/10586458.2015.1005853

Cited by 9 publications

(26 citation statements)

References 48 publications

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“…In [5] a numerical method was used to find bounds for the cogrowth of groups, and a lower bound for the cogrowth rate of BS(N, M ) for small values of N, M were computed. A significant improvement of this numerical work has been undertaken by the authors, which can be found in [4].…”

Section: Introductionmentioning

confidence: 99%

The cogrowth series forBS(N, N) is D-finite

Elder

Rechnitzer

Rensburg

et al. 2014

Int. J. Algebra Comput.

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Communicated by J. MeakinWe compute the cogrowth series for Baumslag-Solitar groups BS(N, N ) = a, b | a N b = ba N , which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.generating function is called the cogrowth series. The rate of exponential growth of the cogrowth function lim sup c(n) 1/n is the cogrowth of the group (with respect to a chosen finite generating set). Note that cogrowth can also be defined by counting only freely reduced trivial words -see Remark 1.1 at the end of this section.In this article we study the cogrowth of the Baumslag-Solitar groupsfor positive integers N, M . We prove in Theorem 4.1 that for groups BS(N, N ) the cogrowth series is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. The class of D-finite (or holonomic) functions includes rational and algebraic functions, and many of the most famous functions in mathematics and physics. See [23,24] for background on D-finite generating functions. If {a n } is a sequence and A(z) = n a n z n is its corresponding generating function then A(z) is D-finite if and only if {a n } is P-recursive (satisfies a linear recurrence with polynomial coefficients). D-finite functions are closed under addition and multiplication, and the composition of D-finite function with an algebraic function is D-finite [23]. Further, if a generating function is D-finite and the differential equation is known, then the coefficients of corresponding sequence can be computed quickly and their asymptotics are readily computed (see for example [26]).The class of group presentations for which the cogrowth series has been computed explicitly (in terms of a closed-form expression, or system of simple recurrences, for example) is limited. Kouksov proved that the cogrowth series is a rational function if and only if the group is finite [15], and computed closed-form expressions for some free products of finite groups and free groups [16] which are algebraic functions. Humphries gave recursions and closed-form functions for various abelian groups [11,12].We note that Dykema and Redelmeier have also tried to compute cogrowth for general Baumslag-Solitar groups [3], and the problem appears to be a difficult one.Grigorchuk and independently Cohen [2,8] proved that a finitely generated group is amenable if and only if its cogrowth rate is twice the number of generators. For more background on amenability and cogrowth see [19,25]. The free group on two (or more) letters is known to be non-amenable, and subgroups of amenable groups are also amenable. It follows that if a group contains a subgroup isomorphic to the free group on two generators, then it cannot be amenable. Z 2 ∼ = BS(1, 1) is amenable, while for N > 1 the subgroup of BS(N, N ) generated by at and at −1 is free, so these groups are non-amenable. We compute cogrowth rates for BS(N, N ) for N ≤ 10 (see Table 1), and observe that the rate appears to converge to that of the free group of rank 2. This is in line with the result of Guyot and Stal...

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Section: Introductionmentioning

confidence: 99%

The cogrowth series forBS(N, N) is D-finite

Elder

Rechnitzer

Rensburg

et al. 2014

Int. J. Algebra Comput.

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“…Extrapolations of our computational results suggest the same as our previous paper, namely, that F might not be amenable. A recent experimental work by Elder, Rechnitzer and Janse van Reusburg [7] on the amenability problem of F using statistical methods arrives also to the same conclusion that F might be non-amenable (see also [8]). As in [10,Section 2], by the norm of ||a|| of an element a in the group ring of a discrete group Γ we mean…”

Section: Introductionmentioning

confidence: 70%

“…(8), (9); the idea is the following: Since (ab) n = Cb(ab) n + C −1 b(ab) n , and b(ab) n is self-adjoint, we can obtain the terms of the expanded sum for C −1 b(ab) n from those of Cb(ab) n by using the "reverse-inverse" map π • J = Ad(D 2 )•R given in Eq. (7). For instance the reverse-inverse of the word CDC 2 D 2 is D 2 R(CDC 2 D 2 ) D 2 = C −1 D −1 C −2 D −2 , which corresponds to reversing the order of the letters and taking the inverse.…”

Section: Resultsmentioning

confidence: 99%

Computational Explorations of the Thompson Group T for the Amenability Problem of F

Haagerup

Ramirez-Solano

2019

Experimental Mathematics

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It is a long standing open problem whether the Thompson group F is an amenable group. In this paper we show that if A, B, C denote the standard generators of Thompson group T and D := CBA −1 thenMoreover, the upper bound is attained if the Thompson group F is amenable.Here, the norm of an element in the group ring CT is computed in B( 2 (T )) via the regular representation of T . Using the "cyclic reduced" numbers τ (((C + C 2 )(D + D 2 + D 3 )) n ), n ∈ N, and some methods from our previous paper [10] we can obtain precise lower bounds as well as good estimates of the spectral distributions of 1 12 ((I+C+C 2 )(I+D+D 2 +D 3 )) * (I+C+C 2 )(I+D+D 2 +D 3 ), where τ is the tracial state on the group von Neumann algebra L(T ). Our extensive numerical computations suggest that 1 √ 12and thus that F might be non-amenable. However, we can in no way rule out that 1 √ 12 ||(I + C + C 2 )(I + D + D 2 + D 3 )|| = 2 + √ 2.

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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%

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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Random Sampling of Trivial Words in Finitely Presented Groups

Cited by 9 publications

References 48 publications

The cogrowth series forBS(N, N) is D-finite

The cogrowth series forBS(N, N) is D-finite

Computational Explorations of the Thompson Group T for the Amenability Problem of F

A computational approach to the Thompson group F

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