Growth of family of finite simple groups (original) (raw)

Milnor's problem on the growth of groups and its consequences

Frontiers in Complex Dynamics, 2014

We present a survey of results related to Milnor's problem on group growth. We discuss the cases of polynomial growth and exponential but not uniformly exponential growth; the main part of the article is devoted to the intermediate (between polynomial and exponential) growth case. A number of related topics (growth of manifolds, amenability, asymptotic behavior of random walks) are considered, and a number of open problems are suggested.

Asymptotic growth and least common multiples in groups

Bulletin of the London Mathematical Society, 2011

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.

On the dimension growth of groups

Journal of Algebra, 2011

We prove that the Thompson group F has exponential dimension growth. We also prove that every solvable finitely generated subgroups of F has polynomial dimension growth while some elementary amenable subgroups of F and some solvable groups of class 3 have dimension growth at least exp( √ n).

On growth and torsion of groups

Groups Geometry and Dynamics, 2009

As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups containing no free subsemigroups on two generators.

The subgroup growth spectrum of virtually free groups

Israel Journal of Mathematics, 2010

For a finitely generated group Γ denote by µ(Γ) the growth coefficient of Γ, that is, the infimum over all real numbers d such that sn(Γ) < n! d. We show that the growth coefficient of a virtually free group is always rational, and that every rational number occurs as growth coefficient of some virtually free group.

On growth of random groups of intermediate growth

Groups, Geometry, and Dynamics, 2014

We study the growth of typical groups from the family of p-groups of intermediate growth constructed by the second author. We find that, in the sense of category, a generic group exhibits oscillating growth with no universal upper bound. At the same time, from a measure-theoretic point of view (i.e., almost surely relative to an appropriately chosen probability measure), the growth function is bounded by e n α for some α < 1.

On the Gap Conjecture concerning group growth

2012

We discuss some new results concerning Gap Conjecture on group growth and present a reduction of it (and its *-version) to several special classes of groups. Namely we show that its validity for the classes of simple groups and residually finite groups will imply the Gap Conjecture in full generality. A similar type reduction holds if the Conjecture is valid for residually polycyclic groups and just-infinite groups. The cases of residually solvable groups and right orderable groups are considered as well.