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.
We exhibit examples of groups of intermediate growth with $2^{\aleph_0}$ ergodic, continuous, invariant random subgroups. The examples are the universal groups associated with a family of groups of intermediate growth.
We construct an uncountable family of finitely generated groups of intermediate growth, with growth functions of new type. These functions can have large oscillations between lower and upper bounds, both of which come from a wide class of functions. In particular, we can have growth oscillating between $e^{n^a}$ and any prescribed function, growing as rapidly as desired. Our construction is built on top of any of the Grigorchuk groups of intermediate growth, and is a variation on the limit of permutational wreath product.
We give a new example of an automata group of intermediate growth. It is generated by an automaton with 4 states on an alphabet with 8 letters. This automata group has exponential activity and its limit space is not simply connected.
We give an example of a 4-regular infinite automatic graph of intermediate growth. It is constructed as a Schreier graph of a certain group generated by 3-state automaton. The question was motivated by an open problem on the existence of Cayley automatic 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.
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.
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.
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…
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.
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).
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…
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.
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 $α<1$.
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 …
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 $g(n) =\log(\log(n)).$
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.
In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} π^{-2} n^{1/4} \exp{π\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$.
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.
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…
We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.
For every infinite sequence $ω=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_ω$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space $\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_ω$ have intermediate growth.
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.
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…
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…
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).
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.
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 $X^3+X^2+X-2$.
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.
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 …
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 …
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.
For an interval exchange map, the number of discontinuities of its iterates either exhibits linear growth or is bounded. This dichotomy is used to prove that the group of interval exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval exchanges. As a further application of this dichotomy, a classification of centralizers in the group of interval exchanges is given. This classification of centralizers is used to compute the automorphism group of the interval exchange group.
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.
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…
