Groups of Intermediate Growth: an Introduction for Beginners

Groups of intermediate growth

I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and derives the growth properties from its analytical behaviour. Unfortunately, there are substantial mistakes in the proofs, so the main claims should be considered unsettled. In particular, it is still an open problem whether the growth of the Gupta-Sidki group is intermediate or exponential.

On FC-central extensions of groups of intermediate growth

It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler and Kassabov-Pak. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to $\mathbb{Z}^{\infty}$; and the other provides groups with prescribed oscillating intermediate growth functions.

Palindromic subshifts and simple periodic groups of intermediate growth

We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group. We show that if the associated Schreier graphs are linearly repetitive, then the group is of intermediate growth. In particular, this gives first examples of simple groups of intermediate growth.

Families of linear semigroups with intermediate growth

Methods from additive number theory are applied to construct families of finitely generated linear semigroups with intermediate growth.

The firefighter problem on polynomial and intermediate growth groups

We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy $\{f(n)\}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment, and give explicit lower bounds depending on the growth rate of the group. These bounds can be further improved when more geometric information is available, such as for Grigo…

On the condensation property of the Lamplighter groups and groups of Intermediate growth

The aim of this short note is to revisit some old results about groups of intermediate growth and groups of the lamplighter type and to show that the Lamplighter group $L=\mathbb{Z}_2\wr \mathbb{Z}$ is a condensation group and has a minimal presentation by generators and relators. The condensation property is achieved by showing that $L$ belongs to a Cantor subset of the space $\mathcal{M}_2$ of marked 2-generated groups consisting mostly of groups of intermediate growth.

On growth of random groups of intermediate growth

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 rigidity of rank gradient in a group of intermediate growth

We introduce and investigate the rigidity property of rank gradient in the case of the group $\mathcal G$ of intermediate growth constructed by the first author. We show that $\mathcal G$ is normally $(f,g)$-RG rigid where $f(n)=\log(n)$ and $g(n) =\log(\log(n)).$

Imbeddings into groups of intermediate growth

Every countable group that does not contain a finitely generated subgroup of exponential growth imbeds in a finitely generated group of subexponential growth. This produces in particular the first examples of groups of subexponential growth containing the additive group of the rationals.

中间增长树的分支数

The branching number of intermediate growth trees

本文引入“中间分支数”（IBN）以刻画中间增长树的分支特性，证明其为树上随机过程的临界阈值，并提出算法在置换圈积中寻找大IBN树，进而首次为某些中间增长群上的消防员问题提供了紧致界。

Aperiodic tilings of manifolds of intermediate growth

We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.

Iterated Monodromy Groups of Intermediate Growth

We give two new examples of groups of intermediate growth, by a method that was first used by Bux and Pérez. Our examples are the groups generated by the automata with the kneading sequences 11(0) and 0(011). By results of Nekrashevych, both of these groups are iterated monodromy groups of complex post-critically finite quadratic polynomials. We include a complete, systematic description of the Bux-Pérez method. We also prove, as a sample application of the method, that the groups determined by the automata with kneading sequence 1(0^k) (k >= 2) have intermediate growth, although this result i…

Distortion of imbeddings of groups of intermediate growth into metric spaces

For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $ρ$, we produce a group of subexponential word growth all of whose imbeddings in $\mathcal X$ have distortion worse than $ρ$. This applies in particular to any B-convex Banach space $\mathcal X$, such as Hilbert space.

No percolation at criticality on certain groups of intermediate growth

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp\left[-Ω(n^γ)\right]$ for some $γ>1/2$. The result is new in the case that the graph is of intermediate volume growth.

Groups of given intermediate word growth

We show that there exists a finitely generated group of growth ~f for all functions f:\mathbb{R}\rightarrow\mathbb{R} satisfying f(2R) \leq f(R)^{2} \leq f(ηR) for all R large enough and η\approx2.4675 the positive root of X^{3}-X^{2}-2X-4. This covers all functions that grow uniformly faster than \exp(R^{\log2/\logη}). We also give a family of self-similar branched groups of growth ~\exp(R^α) for a dense set of α\in(\log2/\logη,1).

On Growth of Generalized Grigorchuk's Overgroups

Grigorchuk's Overgroup $\tilde{\mathcal{G}}$, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group $\mathcal{G}$ of intermediate growth constructed in 1980, but also has elements of infinite order. It's growth is substantially greater than the growth of $\mathcal{G}$. The group $\mathcal{G}$, corresponding to the sequence $(012)^\infty = 012012 ...$, is a member of the family $\{ G_ω| ω\in Ω= \{ 0, 1, 2 \}^\mathbb{N} \}$ consisting of groups of intermediate growth when sequence $ω$ is not virtually constant. Following this construction we define the family…

Nil restricted Lie algebras of oscillating intermediate growth

The Grigorchuk and Gupta-Sidki groups are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. It is known that the famous construction of Golod yields finitely generated associative nil-algebras of exponential growth. Recent extensions of that approach allowed to construct finitely generated associative nil-algebras of polynomial and intermediate growth. Another motivation of the paper is a cons…

Growth of groups with linear Schreier graphs

We introduce a new method of proving upper estimates of growth of finitely generated groups and constructing groups of intermediate growth using graphs of their actions. These estimates are of the form $\exp(n^α)$ for some $α<1$, and provide the first examples of such bounds for simple groups of intermediate growth.

Thue-Morse序列与中间增长群

Thue-Morse sequence and groups of intermediate growth

本文通过嵌入Grigorchuk群到Thue-Morse替换生成的子移位的拓扑全群中，证明了该群包含一个中间增长子群。

虚-nilpotent群中的中间测地生长

Intermediate geodesic growth in virtually nilpotent groups

本文首次提出虚3步nilpotent群具有中间测地生长的例子，并研究了虚nilpotent群的几何性质。

Topological full groups of minimal subshifts with subgroups of intermediate growth

We show that every Grigorchuk group $G_ω$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementary amenable, answering questions of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_ω$ can be embedded in a finitely generated simple group that…

Growth of permutational extensions

We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically $\exp(n^{1-(1-α)^k}),\quad 2^{3-3/α}+2^{2-2/α}+2^{1-1/α}=2$, and a torsion-free group with growth function asymptotically $\exp(\log(n)n^{1-(1-α)^k})$. These are the first examples of groups of intermediate growth for which the growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable …

Substitutional subshifts and growth of groups

We show how to use symbolic dynamics of Schreier graphs to embed the Grigorchuk group into a simple torsion group of intermediate growth and to construct uncountably many growth types of simple torsion groups.

Numerical upper bounds on growth of automata groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automata groups, and more precisely to …

On the Word and Period Growth of some Groups of Tree Automorphisms

We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).

中间增长群的最大子群

Maximal subgroups of groups of intermediate growth

本文解决了作用于有根二叉树的中间增长群的无穷指数最大子群数量问题，证明帐篷映射的非挠迭代单项群恰有可数无穷多个此类子群，并证实Šunić定义的所有群均具有同余子群性质且恰好无穷。

Milnor's Problem on the Growth of Groups and its Consequences

We present a survey of results related to the Milnor's problem on group growth. We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but 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) is considered, and a number of open problems is suggested.

The Growth of Grigorchuk's Group

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth function: $$e^{\sqrt{n}}\precsimγ(n)\precsim e^{n^β},$$ where $β=\log_{32}(31)\approx0.991$. Using elementary methods we bring the upper bound down to $\log(2)/\log(2/η)\approx0.767$, where $η\approx0.811$ is the real root of the polynomial $X^3+X^2+X-2$.

On the growth of iterated monodromy groups

Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of $z^2+i$. This is the first non-trivial example supporting the conjecture.

Lower Bounds on the Growth of Grigorchuk's Torsion Group

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth $γ$: $e^{\sqrt{n}}\precsimγ(n)\precsim e^{n^β},$ where $β=\log_{32}(31)\approx0.991$. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to $e^{n^α}\precsimγ(n),$ where $α\approx0.5157$, with the aid of a computer. This disproves the conjecture that the lower bound be tight.

Conjugacy Growth and Conjugacy Width of Certain Branch Groups

The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further proof that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the …

A Wilson Group of Non-Uniformly Exponential Growth

This note constructs a finitely generated group $W$ whose word-growth is exponential, but for which the infimum of the growth rates over all finite generating sets is 1 -- in other words, of non-uniformly exponential growth. This answers a question by Mikhael Gromov. The construction also yields a group of intermediate growth $V$ that locally resembles $W$ in that (by changing the generating set of $W$) there are isomorphic balls of arbitrarily large radius in $V$ and $W$'s Cayley graphs.

On a family of representations of residually finite groups

For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $γ$ we construct a unitary representation $π_γ$ of $G$. For the minimal growth, $π_γ$ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients $G/G_n$. In the case of intermediate growth we show two examples of different behaviour of $π_γ$.

Exponential growth of some iterated monodromy groups

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the $\operatorname{IMG}$ of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with…