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.
A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-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 $X^3+X^2+X-2$.
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…
On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group $G$ we show that its volume growth function $v_{G,S}(n)$ satisfies that $\lim_{n\to\infty}\log\log v_{G,S}(n)/\log n=α_{0}$, where $α_{0}=\frac{\log2}{\logλ_{0}}\approx0.7674$, $λ_{0}$ is the positive root of the polynomial $X^{3}-X^{2}-2X-4$.
Following the construction from `Degrees of growth of finitely generated groups and the theory of invariant means' we generalize the Grigorchuk's overgroup $\tilde{\mathcal{G}}$, studied in `On parabolic subgroups and Hecke algebras of some fractal groups' to the family $\{ \tilde{G}_ω, ω\in Ω= \{ 0, 1, 2 \}^\mathbb{N} \}$ of generalized Grigorchuk's overgroups. We consider these groups as 8-generated and describe the closure of this family in the space $\mathcal{M}_8$ of marked 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.
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 infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group.
We first show that a class of pro-$p$ branch groups including the Grigorchuk group and the Gupta-Sidki groups all have subgroup growth type $n^{\log n}$. We then introduce the notion of orbit growth and use it to construct extensions of the Grigorchuk group and the Gupta-Sidki groups. We compute the subgroup growth type of these extensions and deduce that all functions between $n^{(\log n)^2}$ and $e^n$ occur as the subgroup growth type of a pro-$p$ group, thus, giving almost complete answer to a question raised by Lubotzky and Segal.
We address a question of Grigorchuk by providing both a system of recursive formulas and an asymptotic result for the portrait growth of the first Grigorchuk group. The results are obtained through analysis of some features of the branching subgroup structure of the group. More generally, we provide recursive formulas for the portrait growth of any finitely generated, contracting, regular branch group, based on the coset decomposition of the groups that are higher in the branching subgroup structure in terms of the lower subgroups. Using the same general approach we fully describe the portrait…
We study the subgroup structure of the infinite torsion $p$-groups defined by Gupta and Sidki in 1983. In particular, following results of Grigorchuk and Wilson for the first Grigorchuk group, we show that all infinite finitely generated subgroups of the Gupta--Sidki 3-group $G$ are abstractly commensurable with $G$ or $G\times G$. As a consequence, we show that $G$ is subgroup separable and from this it follows that its membership problem is soluble. Along the way, we obtain a characterization of finite subgroups of $G$ and establish an analogue for the Grigorchuk group.
Fixing a subgroup $Γ$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ of $G$ with $[Γ: Γ\cap Δ][Δ: Γ\cap Δ] = n$. For pairs $Γ\leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Γ$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Γ$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $\aleph_0$ for any $n …
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…
There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-menti…
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).
Grigorchuk and de la Harpe asked if there are many groups with growth exponent close to that of the free group with the same number of generators. We prove that this is in fact the case for a generic group (in the density model of random groups). More generally, we prove that the growth exponent does not change much through a random quotient of a torsion-free hyperbolic group.
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.
In this note we describe the role of the Schur multiplier in the structure of the $p$-torsion of discrete groups. More concretely, we show how the knowledge of $H_2G$ allows to approximate many groups by colimits of copies of finite $p$-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable examples and the first Grigorchuk group. We also provide a counterexample for a conjecture of E. Farjoun.
We prove the exponential growth of product replacement graphs for a large class of groups. Much of our effort is dedicated to the study of product replacement graphs of Grigorchuk groups, where the problem is most difficult.
A function $\mathbb{N} \to \mathbb{N}$ is near exponential if it is bounded above and below by functions of the form $2^{n^c}$ for some $c > 0$. In this article we develop tools to recognize the near exponential residual finiteness growth in groups acting on rooted trees. In particular, we show the near exponential residual finiteness growth for certain branch groups, including the first Grigorchuk group, the family of Gupta-Sidki groups and their variations, and Fabrykowski-Gupta groups. We also show that the family of Gupta-Sidki p-groups, for $p\geq 5$, have super-exponential residual finit…
We prove that the so called Grigorchuk-Maki group of intermadiate growth can be seen as a group of $C^1$ diffeomorphisms of the interval. On the other hand, we prove that every group of $C^{1+α}$ diffeomorphisms of the interval having subexponential growth is almost nilpotent.
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 …
We prove that torsion in the abelianizations of open normal subgroups in finitely presented pro-$p$ groups can grow arbitrarily fast. By way of contrast in $\mathbb Z_p$- analytic groups the torsion growth is at most polynomial.
We study the growth of torsion in the abelianizations of finite index subgroups in finitely generated metabelian groups. This complements earlier work of Kar, Kropholler and the author which covered the finitely presented amenable groups.
We study completions of the group algebra of a finitely generated group and relate nuclearity of such a completion to growth properties of the group. This extends previous work of Jolissaint on nuclearity of rapidly decreasing functions on a finitely generated group to more general weights than polynomial decrease. The new group algebras and their duals are studied in detail and compared to other approaches. As application we discuss the convergence of the complete growth function introduced by Grigorchuk and Nagnibeda.
We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups containing no free subsemigroups on two generators.
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…
We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call "parabolic space", namely the quotient of the group by the stabilizer of an infinite ray. The growth of th…
We show that Lück's conjecture on torsion growth in homology implies that two 3-manifolds have equal volume if the fundamental groups have the same set of finite quotients.
A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight. Examples of non-growth tight actions are product groups acting on the $L^1$ products of Cayley graphs of the factors. In this paper we consider actions of product groups on produc…
We show that finitely generated Abelian groups admit equal growth functions with respect to symmetric generating sets if and only if they have the same rank and the torsion parts have the same parity. In contrast, finitely generated Abelian groups admit equal growth functions with respect to monoid generating sets if and only if they have same rank. Moreover, we show that the size of the torsion part is in fact determined by the set of all growth functions of a finitely generated Abelian group.
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 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 show that every Coxeter group that is not virtually abelian and for which all labels in the corresponding Coxeter graph are powers of 2 or infinity can be mapped onto uncountably many infinite 2-groups which, in addition, may be chosen to be just-infinite, branch groups of intermediate growth. Also we answer affirmatively a question raised by Wiegold in Kourovka Notebook.
Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem…
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_Γ : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_Γ$ is bounded iff $Γ$ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function $f: \mathbb{N} \rightarrow \mathbb{N}$, an elementary amenable lawless groups for which $\mathcal{A}_Γ$ grows more slowly that $f$. We produce torsion lawless groups for which $\mathcal{A}_Γ$ is at least linear using Golod-Shafarevich theory, and give some upper bounds on $…
Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ…
Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free 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 …
Let G be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient G/G^n tends to the one of G as n odd approaches infinity. Moreover we provide an estimate at which the convergence is taking place.
In this paper, we determine the descriptive complexity of subsets of the Polish space of marked groups defined by various group theoretic properties. In particular, using Grigorchuk groups, we establish that the sets of solvable groups, groups of exponential growth and groups with decidable word problem are $\mathbfΣ^0_2$-complete and that the sets of periodic groups and groups of intermediate growth are $\mathbfΠ^0_2$-complete. We also provide bounds for the descriptive complexity of simplicity, amenability, residually finiteness, Hopficity and co-Hopficity. This paper is intended to serve as…
Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume $H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-co…
