Subgroup Growth

Lubotzky, Alexander; Segal, David R.

doi:10.1007/978-3-0348-8965-0

Cited by 273 publications

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“…The class of virtually solvable groups with finite rank is known to have polynomial subgroup growth (see [9,Chapter 5]) and thus have a polynomial upper bound on normal subgroup growth. Using this upper bound with (1) yields our next result.…”

Section: Proposition 24 Let γ Be a Finitely Generated Residually Fmentioning

confidence: 99%

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Asymptotic growth and least common multiples in groups

Bou-Rabee¹,

McReynolds²

2011

Bulletin of the London Mathematical Society

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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.

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Section: Proposition 24 Let γ Be a Finitely Generated Residually Fmentioning

confidence: 99%

“…Our next result shows that residual girth functions enjoy the same growth dichotomy as word and subgroup growth-see [5] and [9]. Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic growth and least common multiples in groups

Bou-Rabee¹,

McReynolds²

2011

Bulletin of the London Mathematical Society

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“…(a) follows from [LS,Proposition 1.6.2] while (b) follows from Proposition 4.4 above. Now, by applying (a) and then (b) we have…”

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confidence: 96%

Representation growth of linear groups

Larsen¹,

Lubotzky²

2008

J. Eur. Math. Soc.

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110

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Abstract. Let be a group and r n ( ) the number of its n-dimensional irreducible complex representations. We define and study the associated representation zeta function Z (s) = ∞ n=1 r n ( )n −s . When is an arithmetic group satisfying the congruence subgroup property then Z (s) has an "Euler factorization". The "factor at infinity" is sometimes called the "Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups U of the associated simple group G over the associated local field K. Here we show a surprising dichotomy: if G(K) is compact (i.e. G anisotropic over K) the abscissa of convergence goes to 0 when dim G goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.

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“…Later a classification free proof was given by Nori [Nor87] using methods of algebraic geometry. Other treatments of the theorem can be found in [LS03], [HP95], [Pin00]. The following follows directly by applying the theorem in the case n = 2, and recalling that the only subgroups of PSL 2 (Z) that fail to be Zariski dense are cyclic groups:…”

Section: Connectednessmentioning

confidence: 99%

Strong approximation in random towers of graphs

Glasner

2014

Combinatorica

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Abstract. The term strong approximation is used to describe phenomena where an arithmetic group as well as all of its Zariski dense subgroups have a large image in the congruence quotients. We exhibit analogues of such phenomena in a probabilistic, rather than arithmetic, setting.Let T be the binary rooted tree, Aut(T ) its automorphism group. To a given m-tuple a = {a 1 , a 2 , . . . , a m } ∈ Aut(T ) m we associate a tower of 2m-regular Schreier graphs . . . → X n → X n−1 → . . . → X 0 .

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Subgroup Growth

Cited by 273 publications

References 0 publications

Asymptotic growth and least common multiples in groups

Asymptotic growth and least common multiples in groups

Representation growth of linear groups

Strong approximation in random towers of graphs

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