Quantifying residual finiteness

Bou-Rabee, Khalid

doi:10.1016/j.jalgebra.2009.10.008

Cited by 63 publications

(151 citation statements)

References 4 publications

Supporting

Mentioning

151

Contrasting

Order By: Relevance

“…It was established in [1] that log(n) max D Γ,X (n) for any finitely generated group with an element of infinite order (this was also shown by [11]). For a nonabelian free group F m of rank m, Bogopolski asked whether max D F m ,X (n) ≃ log(n) (see Problem 15.35 in the Kourovka notebook [10]).…”

Section: Introductionmentioning

confidence: 75%

“…In Section 4, we extend this approach by generalizing least common multiples to finitely generated groups (a similar approach was also taken in the article of Hadad [7]). Indeed with this analogy, Theorem 1.2 and the upper bound of n 3 established in [1], [11] can be viewed as a weak Prime Number Theorem for free groups since the Prime Number Theorem yields F Z (n) ≃ log(n). Recently, Kassabov-Matucci [8] improved the lower bound of n 1/3 to n 2/3 .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic growth and least common multiples in groups

Bou-Rabee¹,

McReynolds²

2011

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

In this article we relate word and subgroup growth to certain functions that arise in the quantification of residual finiteness. One consequence of this endeavor is a pair of results that equate the nilpotency of a finitely generated group with the asymptotic behavior of these functions. The second half of this article investigates the asymptotic behavior of two of these functions. Our main result in this arena resolves a question of Bogopolski from the Kourovka notebook concerning lower bounds of one of these functions for nonabelian free groups.

show abstract

Section: Introductionmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Asymptotic growth and least common multiples in groups

Bou-Rabee¹,

McReynolds²

2011

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…The full residual finiteness growth of the discrete Heisenberg group is presented in [8]. Compare full residual finiteness growth to the concept of residual finiteness growth, which measures how well individual elements are detected by finite quotients, appearing in [5], [7], [3], [11], [18], [6], [12]. Also compare this with Sarah Black's growth function defined and studied in [2].…”

Section: Theorem 3 Let G Be a Finitely Generated Nilpotent Group Thmentioning

confidence: 99%

Full residual finiteness growths of nilpotent groups

Bou-Rabee

Studenmund

2016

Isr. J. Math.

Self Cite

View full text Add to dashboard Cite

Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely n b , where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form n b where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.

show abstract

“…In fact one can have much smaller bounds for many linear groups. For example, for the free group F 2 , ρ F 2 (n) n 3 by [9]. A lower bound for the depth function for a free group is equivalent to n 2 virtually residually nilpotent (proved by Borisov and the third author [6,7]) but the only upper bound one can deduce from the proof is exponential.…”

Section: The Time Function Of the Algorithm A No : The Depth Functionmentioning

confidence: 99%