Groups of Oscillating Intermediate Growth
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Arxiv:1701.04633V1 [Math.NT]
COUNTING IDEALS IN POLYNOMIAL RINGS LENNY FUKSHANSKY, STEFAN KUHNLEIN,¨ AND REBECCA SCHWERDT Abstract. We investigate properties of zeta functions of polynomial rings and their quotients, generalizing and extending some classical results about Dedekind zeta functions of number fields. By an application of Delange’s version of the Ikehara Tauberian Theorem, we are then able to determine the asymptotic order of the ideal counting function in such rings. As a result, we produce counting estimates on ideal lattices of bounded determinant coming from fixed number fields, as well as density estimates for any ideal lattices among all sublattices of Zd. We conclude with some more general speculations and open questions. 1. Introduction A classical arithmetic problem in the theory of finitely generated groups and rings is the study of the asymptotic order of growth of the number of subgroups of bounded index. A common approach to this problem involves studying the analytic properties of a corresponding zeta function (a Dirichlet series generating function) and then applying a Tauberian theorem to deduce information about the number in question, represented by the coefficients of this zeta function. This research direction received a great deal of attention over the years as can be seen from [11], [8], [6], [7], [3], [17] and the references within. In the recent years, a similar approach has also been applied to the more geometric setting of counting sublattices in lattices, e.g. [15], [10], [9], [1], [14]. In this note, we consider some special cases of the following general setting. Let R be a commutative ring with identity such that for every natural number n the set of ideals in R of index n is a finite number, call this number an(R). -
On Just-Infiniteness of Locally Finite Groups and Their -Algebras
Bull. Math. Sci. DOI 10.1007/s13373-016-0091-4 On just-infiniteness of locally finite groups and their C∗-algebras V. Belyaev1 · R. Grigorchuk2 · P. Shumyatsky3 Received: 14 August 2016 / Accepted: 23 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We give a construction of a family of locally finite residually finite groups with just-infinite C∗-algebra. This answers a question from Grigorchuk et al. (Just- infinite C∗-algebras. https://arxiv.org/abs/1604.08774, 2016). Additionally, we show that residually finite groups of finite exponent are never just-infinite. Keywords Just infinite groups · C*-algebras Mathematics Subject Classification 46L05 · 20F50 · 16S34 For Alex Lubotzky on his 60th birthday. Communicated by Efim Zelmanov. R. Grigorchuk was supported by NSA Grant H98230-15-1-0328. P. Shumyatsky was supported by FAPDF and CNPq. B P. Shumyatsky [email protected] V. Belyaev [email protected] R. Grigorchuk [email protected] 1 Institute of Mathematics and Mechanics, S. Kovalevskaja 16, Ekaterinburg, Russia 2 Department of Mathematics, Texas A&M University, College Station, TX, USA 3 Department of Mathematics, University of Brasília, 70910 Brasília, DF, Brazil 123 V. Belyaev et al. 1 Introduction A group is called just-infinite if it is infinite but every proper quotient is finite. Any infi- nite finitely generated group has just-infinite quotient. Therefore any question about existence of an infinite finitely generated group with certain property which is pre- served under homomorphic images can be reduced to a similar question in the class of just-infinite groups. -
Arxiv:1510.00545V4 [Math.GR] 29 Jun 2016 7
SCHREIER GRAPHS OF GRIGORCHUK'S GROUP AND A SUBSHIFT ASSOCIATED TO A NON-PRIMITIVE SUBSTITUTION ROSTISLAV GRIGORCHUK, DANIEL LENZ, AND TATIANA NAGNIBEDA Abstract. There is a recently discovered connection between the spectral theory of Schr¨o- 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-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators. Contents Introduction 2 1. Subshifts and aperiodic order in one dimension 3 2. Schr¨odingeroperators with aperiodic order 5 2.1. Constancy of the spectrum and the integrated density of states (IDS) 5 2.2. The spectrum as a set and the absolute continuity of spectral measures 9 2.3. Aperiodic order and discrete random Schr¨odingeroperators 10 3. The substitution τ, its finite words Subτ and its subshift (Ωτ ;T ) 11 3.1. The substitution τ and its subshift: basic features 11 3.2. The main ingredient for our further analysis: n-partition and n-decomposition 14 3.3. The maximal equicontinuous factor of the dynamical system (Ωτ ;T ) 15 3.4. Powers and the index (critical exponent) of Subτ 19 3.5. -
What Is the Grigorchuk Group?
WHAT IS THE GRIGORCHUK GROUP? JAKE HURYN Abstract. The Grigorchuk group was first constructed in 1980 by Rostislav Grigorchuk, defined as a set of measure-preserving maps on the unit interval. In this talk a simpler construction in terms of binary trees will be given, and some important properties of this group will be explored. In particular, we study it as a negative example to a variant of the Burnside problem posed in 1902, an example of a non-linear group, and the first discovered example of a group of intermediate growth. 1. The Infinite Binary Tree We will denote the infinite binary tree by T . The vertex set of T is all finite words in the alphabet f0; 1g, and two words have an edge between them if and only if deleting the rightmost letter of one of them yields the other: ? 0 1 00 01 10 11 . Figure 1. The infinite binary tree T . This will in fact be a rooted tree, with the root at the empty sequence, so that any automorphism of T must fix the empty sequence. Here are some exercises about Aut(T ), the group of automorphisms of T . In this paper, exercises are marked with either ∗ or ∗∗ based on difficulty. Most are taken from exercises or statements made in the referenced books. Exercise 1. Show that the group Aut(T ) is uncountable and has a subgroup iso- morphic to Aut(T ) × Aut(T ).∗ Exercise 2. Impose a topology on Aut(T ) to make it into a topological group home- omorphic to the Cantor set.∗∗ Exercise 3. -
The Commensurability Relation for Finitely Generated Groups
J. Group Theory 12 (2009), 901–909 Journal of Group Theory DOI 10.1515/JGT.2009.022 ( de Gruyter 2009 The commensurability relation for finitely generated groups Simon Thomas (Communicated by A. V. Borovik) Abstract. There does not exist a Borel way of selecting an isomorphism class within each com- mensurability class of finitely generated groups. 1 Introduction Two groups G1, G2 are (abstractly) commensurable, written G1 QC G2, if and only if there exist subgroups Hi c Gi of finite index such that H1 G H2. In this paper, we shall consider the commensurability relation on the space G of finitely generated groups. Here G denotes the Polish space of finitely generated groups introduced by Grigorchuk [4]; i.e., the elements of G are the isomorphism types of marked groups 3G; c4, where G is a finitely generated group and c is a finite sequence of generators. (For a clear account of the basic properties of the space G, see either Champetier [1] or Grigorchuk [6].) Our starting point is the following result, which intuitively says that the isomorphism and commensurability relations on G have exactly the same complexity. Theorem 1.1. The isomorphism relation G and commensurability relation QC on the space G of finitely generated groups are Borel bireducible. In particular, there exists a Borel reduction from QC to G; i.e., a Borel map f : G ! G such that if G, H a G, then G QC H if and only if f ðGÞ G f ðHÞ: However, while the proof of Theorem 1.1 shows that there exists a Borel reduction from QC to G, it does not provide an explicit example of such a reduction. -
Groups Acting on 1-Dimensional Spaces and Some Questions
GROUPS ACTING ON 1-DIMENSIONAL SPACES AND SOME QUESTIONS Andr´esNavas Universidad de Santiago de Chile ICM Special Session \Dynamical Systems and Ordinary Differential Equations" Rio de Janeiro, August 2018 A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Vertices: elements of the group. Edges: connect any two elements that differ by (right) multiplication by a generator. Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). A group is a set endowed with a multiplication and an inversion sa- tisfying certain formal rules/axioms (Galois, Cayley). Theorem (Cayley) Every group is a subgroup of the group of automorphisms of a cer- tain space (the group itself / its Cayley graph). Groups \Les math´ematiquesne sont qu'une histoire de groupes" (Poincar´e). -
Arithmetic Lattices in Unipotent Algebraic Groups 2
Arithmetic lattices in unipotent algebraic groups 1 Arithmetic lattices in unipotent algebraic groups Khalid Bou-Rabee∗ Daniel Studenmund† April 19, 2018 Abstract Fixing an arithmetic lattice Γ in an algebraic group G, the commensurability growth function assigns to each n the cardinality of the set of subgroups ∆ with [Γ : Γ ∩ ∆][∆ : Γ ∩ ∆] = n. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith’s “Subgroups of finite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic Z-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in p−s, where the degrees of the numerator and denominator are independent of p. This gives regularity results for the set of arithmetic lattices in G. keywords: commensurability, lattices, unipotent algebraic groups, subgroup growth 1 Introduction Let G be an algebraic group defined over Z (an algebraic Z-group). Two subgroups ∆1 and ∆2 of G(R) are commensurable if their commensurability index c(∆1,∆2) := [∆1 : ∆1 ∩ ∆2][∆2 : ∆1 ∩ ∆2] is finite. An arithmetic lattice of G is a subgroup of G(R) that is commensurable with G(Z). The first purpose of this article is to show that when G is unipotent, the set of arithmetic lattices in G has a great deal of regularity. The second purpose is to bring attention to a new notion of quantifying commensurability. The main tool of this article is the commensurability growth function N → N ∪{∞} assigning to arXiv:1804.04973v2 [math.GR] 18 Apr 2018 each n ∈ N the cardinality cn(G(Z),G(R)) := |{∆ ≤ G(R) :c(G(Z),∆)= n}|. -
THE GRIGORCHUK GROUP Contents Introduction 1 1. Preliminaries 2 2
THE GRIGORCHUK GROUP KATIE WADDLE Abstract. In this survey we will define the Grigorchuk group and prove some of its properties. We will show that the Grigorchuk group is finitely generated but infinite. We will also show that the Grigorchuk group is a 2-group, meaning that every element has finite order a power of two. This, along with Burnside's Theorem, gives that the Grigorchuk group is not linear. Contents Introduction 1 1. Preliminaries 2 2. The Definition of the Grigorchuk Group 5 3. The Grigorchuk Group is a 2-group 8 References 10 Introduction In a 1980 paper [2] Rostislav Grigorchuk constructed the Grigorchuk group, also known as the first Grigorchuk group. In 1984 [3] the group was proved by Grigorchuk to have intermediate word growth. This was the first finitely generated group proven to show such growth, answering the question posed by John Milnor [5] of whether such a group existed. The Grigorchuk group is one of the most important examples in geometric group theory as it exhibits a number of other interesting properties as well, including amenability and the characteristic of being just-infinite. In this paper I will prove some basic facts about the Grigorchuk group. The Grigorchuk group is a subgroup of the automorphism group of the binary tree T , which we will call Aut(T ). Section 1 will explore Aut(T ). The group Aut(T ) does not share all of its properties with the Grigorchuk group. For example we will prove: Proposition 0.1. Aut(T ) is uncountable. The Grigorchuk group, being finitely generated, cannot be uncountable. -
Subgroup Growth: an Introduction
Subgroup growth: An introduction Ashok Rajaraman 301121276 [email protected] November 30, 2011 Abstract A major topic in geometric group theory is the counting the number of subgroups of finite index in a group. This ties into the classification of finite simple groups, as well as classical geometric group theory, to culminate in a theorem similar in spirit to Gromov's theorem on polynomial growth of finitely generated groups. This short paper captures some of the elementary results in this field. 1 Introduction Let us associate the following function with a group G: n 7! an (G) ; where an (G) is a natural number which denotes the number of subgroups of index n in G. This is the subgroup growth function of G [1]. This function is well defined if the value of an (G) is finite for all values of n 2 N. The study of the behaviour of this function was motivated by the need to classify infinite groups by some invariant. Infinite groups with the same `type' of subgroup growth are expected to show similar properties, as we shall see later on. In turn, subgroup growth has been the motivation for some major fields in group theory, such as strong approximation and linearity conditions for linear groups. The completion of the classification of finite simple groups proved a major turning point in the field, and a comprehensive theory of subgroup growth has been formulated by the works of Lubotzky, Segal, Mann [2], Larsen [3] and Ilani [4], to name a few. However, the tools used to analyze subgroup growth date much farther into the past. -
On Growth of Grigorchuk Groups
On growth of Grigorchuk groups Roman Muchnik, Igor Pak March 16, 1999 Abstract We present an analytic technique for estimating the growth for groups of intermediate growth. We apply our technique to Grigorchuk groups, which are the only known examples of such groups. Our esti- mates generalize and improve various bounds by Grigorchuk, Bartholdi and others. 2000 Mathematics Subject Classification: 20E08, 20E69, 68R15 1 Introduction In a pioneer paper [3] R. Grigorchuk discovered a family of groups of inter- mediate growth, which gave a counterexample to Milnor's Conjecture (see [3, 7, 13]). The groups are defined as groups of Lebesgue-measure-preserving transformations on the unit interval, but can be also defined as groups act- ing on binary trees, by finite automata, etc. While Grigorchuk was able to find both lower and upper bounds on growth, there is a wide gap between them, and more progress is desired. In this paper we present a unified approach to the problem of estimating the growth. We introduce an analytic result we call Growth Theorem, which lies in the heart of our computations. This reduces the problem to combi- natorics of words which is a natural language in this setting. We proceed to obtain both upper and lower bounds in several cases. This technique simplifies and improves the previous bounds obtained by various ad hoc ap- proaches (see [2, 4, 5]). We believe that our Growth Theorem can be also applied to other classes of groups. Let G be an infinite group generated by a finite set S, S = S−1, and let Γ be the corresponding Cayley graph. -
Arxiv:1808.06371V2 [Math.GR] 11 Sep 2019 N Ugop N H Oe Rwhsre Ihrsett N Su Set
RATIONAL GROWTH IN VIRTUALLY ABELIAN GROUPS ALEX EVETTS Abstract. We show that any subgroup of a finitely generated virtu- ally abelian group G grows rationally relative to G, that the set of right cosets of any subgroup of G grows rationally, and that the set of conju- gacy classes of G grows rationally. These results hold regardless of the choice of finite weighted generating set for G. Key words: Conjugacy growth, relative growth, coset growth, virtu- ally abelian groups. 1. Introduction The related notions of growth functions and growth series of finitely gen- erated groups have attracted a lot of attention in many different classes of groups [14]. In the case of growth series, it is perhaps surprising that there are very few results that are independent of the choice of finite generating set. In this paper we add three facts to those results. Namely that for a virtu- ally abelian group, the conjugacy growth series, the relative growth series of any subgroup, and the coset growth series with respect to any subgroup, are rational functions for any choice of generating set. The standard weighted growth series of any virtually abelian group was shown by Benson to be rational in [3]. Liardet [23] built on work of Klarner [22] and others to show that the complete growth series (a generalisation of standard growth) is also rational. The corresponding result (with weight function uniformly equal to 1) was proved for hyperbolic groups in the 1980s (see [5], [6], [16], and [18]), and for the integer Heisenberg group in [15]. The significance of these results comes from the fact that they hold for any choice of generating set S. -
Groups of Intermediate Growth and Grigorchuk's Group
Groups of Intermediate Growth and Grigorchuk's Group Eilidh McKemmie supervised by Panos Papazoglou This was submitted as a 2CD Dissertation for FHS Mathematics and Computer Science, Oxford University, Hilary Term 2015 Abstract In 1983, Rostislav Grigorchuk [6] discovered the first known example of a group of intermediate growth. We construct Grigorchuk's group and show that it is an infinite 2-group. We also show that it has intermediate growth, and discuss bounds on the growth. Acknowledgements I would like to thank my supervisor Panos Papazoglou for his guidance and support, and Rostislav Grigorchuk for sending me copies of his papers. ii Contents 1 Motivation for studying group growth 1 2 Definitions and useful facts about group growth 2 3 Definition of Grigorchuk's group 5 4 Some properties of Grigorchuk's group 9 5 Grigorchuk's group has intermediate growth 21 5.1 The growth is not polynomial . 21 5.2 The growth is not exponential . 25 6 Bounding the growth of Grigorchuk's group 32 6.1 Lower bound . 32 6.1.1 Discussion of a possible idea for improving the lower bound . 39 6.2 Upper bound . 41 7 Concluding Remarks 42 iii iv 1 Motivation for studying group growth Given a finitely generated group G with finite generating set S , we can see G as a set of words over the alphabet S . Defining a weight on the elements of S , we can assign a length to every group element g which is the minimal possible sum of the weights of letters in a word which represents g.