DOCSLIB.ORG

Arithmetic Groups, Base Change, and Representation Growth

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Arithmetic Groups, Base Change, and Representation Growth ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Abstract. Consider an arithmetic group G(OS), where G is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers OS of a number field K with respect to a finite set of places S. For each n 2 N, let Rn(G(OS)) denote the number of irreducible complex representations of G(OS) of dimension at most n. The degree of representation growth α(G(OS)) = limn!1 log Rn(G(OS))= log n is finite if and only if G(OS) has the weak Congruence Subgroup Property. We establish that for every G(OS) with the weak Congruence Subgroup Property the invariant α(G(OS)) is already determined by the absolute root system of G. To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions K ⊂ L. We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky's conjecture to Serre's conjecture on the weak Congruence Subgroup Property, which it refines. Contents 1. Introduction and Main Results 2 1.1. Background and Motivation 2 1.2. Discussion of Main Results 3 2. Reduction of the Main Results to Theorem 2.8 7 3. Base Change for Finite Groups of Lie Type 14 3.1. Finite Groups of Lie Type 14 3.2. Applications to Finite Quotients of Arithmetic Groups 26 4. Relative Zeta Functions, Kirillov Orbit Method and Model Theoretic Background 27 4.1. Relative Zeta Functions and Cohomology 27 4.2. Kirillov Orbit Method 29 4.3. Quantifier-Free Definable Sets and Functions 32 4.4. Valued Fields 38 5. Parametrizing Representations 40 5.1. Relative Orbit Method 40 5.2. The Stabilizer of ΞR 43 5.3. The Lie Algebra Associated to the Stabilizer of ΞR 48 2 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 5.4. Proof of Theorem 5.1 51 6. Quantifier-Free Integrals 56 6.1. Uniform Formulae for Quantifier-Free Integrals 56 6.2. Proof of Theorem 6.2 57 6.3. Proof of Theorem 2.8 63 References 64 1. Introduction and Main Results 1.1. Background and Motivation. One of the aims of this paper is to prove a variant of a conjecture of Larsen and Lubotzky on the representation growth of irreducible lattices in higher rank semi-simple groups. We recall that the representation growth of an arbitrary group G is given by the asymptotic behavior of the sequence Rn(G), n 2 N, where Rn(G) denotes the number of equivalence classes of irreducible complex representations of G of dimension at most n. Whenever G is a topological (resp. algebraic) group, we restrict the investigation without further comment to continuous (resp. rational) representations. Ac- cording to Margulis' Arithmeticity Theorem, the lattices in question arise in the following way. Consider an arithmetic group G(OS), where OS is the ring of S-integers of a number field K with respect to a finite set of places S of K and G is an affine group scheme over OS whose generic fiber is connected, simply connected absolutely almost simple. Suppose P further that the S-rank of G, i.e., rkS G = v2S rkKv G, is at least 2 and that an infinite place v is included in S if rkKv G ≥ 1. A theorem of Borel and Harish{Chandra shows that the image of G(OS) under the diagonal embedding is indeed an irreducible lattice Q in the higher rank semi-simple group H = v2S G(Kv). Moreover, Margulis proved that this construction produces, up to commensurability, essentially all irreducible lattices in higher rank semi-simple groups. Precise notions and a more complete description can be found in [33]. In this paper, we call a group arithmetic if it is commensurable to a group of the form G(OS) as above. In particular, all arithmetic groups that we consider are defined in characteristic 0. The study of representation growth for arithmetic groups was initiated by Lubotzky and Martin in [30]. They showed that, whenever Γ is commensurable to G(OS) as above and rkKp G ≥ 1 for every finite place p 2 S, then the growth of the sequence Rn(Γ), n 2 N, is bounded polynomially in n if and only if G(OS) has the weak Congruence Subgroup Property. To discuss the latter, let G\(OS) and OcS denote the profinite completions of the group G(OS) and the ring OS. Furthermore, we write Op for the completion of the ring of integers O of K at a prime p. The group G(OS) has the weak Congruence Subgroup Property (wCSP) if the kernel of the natural map ∼ Y G\(OS) −! G(OcS) = G(Op); p2Spec(O)rS ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 3 is finite. A long-standing conjecture of Serre asserts, in particular, that G(OS) has the wCSP whenever rkS G ≥ 2 and rkKp G ≥ 1 for every finite place p 2 S. This part of Serre's conjecture is known to be true in many cases; e.g., it holds for groups yielding non-uniform irreducible lattices in higher rank semi-simple groups. For more information see [37, Chapter 9.5], [39] or [38], and the references therein. Next we recall the definition of the representation zeta function of a group G and its abscissa of convergence, which captures the degree of representation growth of G. Definition. Let G be a group such that Rn(G) is finite for all n 2 N. The representation zeta function of G is the Dirichlet generating series X −s ζG(s) = (dim %) ; %2Irr(G) where Irr(G) is the set of equivalence classes of finite-dimensional irreducible complex representations % of G and s 2 C is a complex variable. The abscissa of convergence of ζG(s) is the infimum of all σ 2 R such that the series ζG(s) converges absolutely for all s 2 C with Re(s) > σ; we denote this invariant by α(G). In particular, α(G) = 1 if ζG(s) diverges for all s 2 C. Whenever a group G, as in the definition above, possesses infinitely many finite- dimensional irreducible complex representations, the abscissa of convergence α(G) is re- lated to the asymptotic behavior of the sequence Rn(G) by the equation log R (G) (1.1) α(G) = lim sup n : n!1 log n In the case of an arithmetic group Γ = G(OS), as described above, Lubotzky and Martin's result in [30] can therefore be stated as follows: Γ has the wCSP if and only if α(Γ) < 1. In this sense the invariant α(Γ) provides a means to study the wCSP in a quantitative log Rn(Γ) way. We also remark that if Γ has the wCSP, then α(Γ) = limn!1 log n is actually a α(Γ)+o(1) limit, hence Rn(Γ) = n ; this is shown implicitly in [2]. 1.2. Discussion of Main Results. In this paper we establish new quantitative results regarding the representation growth of arithmetic groups with the wCSP. Our first main theorem is the following. Theorem 1.1. Let Φ be an irreducible root system. Then there exists a constant αΦ such that, for every arithmetic group G(OS), where OS is the ring of S-integers of a number field K with respect to a finite set of places S and G is an affine group scheme over OS whose generic fiber is connected, simply connected absolutely almost simple with absolute root system Φ, the following holds: if G(OS) has the wCSP, then α(G(OS)) = αΦ. The theorem highlights two challenging open problems, namely to determine the con- stants αΦ and to establish finer asymptotics for the representation growth of arithmetic groups with the wCSP. Even at the conjectural level we are presently very far from solv- ing these problems. The main theorem in [2] shows that αΦ 2 Q for all Φ. Furthermore, 4 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 1 αΦ ≥ 15 (see [28, Theorem 8.1]) and αA` ≤ 22 for all ` 2 N, with similar bounds on other root systems (see [1]). The only precisely known values are αA1 = 2 (see [28, The- orem 10.1]) and αA2 = 1 (see [3, Theorem C]). In fact, for arithmetic groups Γ = G(OS) with the wCSP which arise from affine group schemes G of type A1 or A2, even finer asymptotics of the representation growth of Γ have been established. If G has abso- lute root system A1, then ζΓ(s) admits a meromorphic continuation beyond its abscissa of convergence and has a simple pole at s = 2 (compare [28] and [4]); consequently, 2 Rn(Γ) = (cΓ + o(1))n for a constant cΓ 2 R. Similarly, if G has absolute root system A2, then ζΓ(s) has a meromorphic continuation beyond its abscissa of convergence and a double pole at s = 1 (see [4]); consequently, Rn(Γ) = (cΓ + o(1))n log n for a constant cΓ 2 R. For general Γ, it remains open whether and how far ζΓ(s) can be extended mero- morphically and, if so, whether the order of the resulting pole at s = α(Γ) depends only on the absolute root system Φ.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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}|.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Relative Subgroup Growth and Subgroup Distortion
    Relative Subgroup Growth and Subgroup Distortion Tara C. Davis, Alexander Yu. Olshanskii October 29, 2018 Abstract We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embed- dings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generted subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group. 1 Introduction and Background We will use the notation that for a group G with finite generating set X, and for an element g G, then g X represents the word length of ∈ | | the element g with respect to the generating set X. For this paper, N = 1, 2,..., . For r N we use the notation that BG(r) represents the ball { } ∈ in G centered at the identity with radius r. That is, BG(r) = g G : { ∈ g X r . | | ≤ } Definition 1.1. Let G be a group finitely generated by a set X with any subset H. The growth function of H in G is G g : N N : r # h H : h X r = #(BG(r) H). H → → { ∈ | | ≤ } ∩ arXiv:1212.5208v1 [math.GR] 20 Dec 2012 The growth functions of subsets and other functions of this type be- come asymptotic invariants independent of the choice of a finite set X if one takes them up to a certain equivalence (see the definitions of , , ∼ ∼ and in Section 3.) ≈ The growth of a subgroup H with respect to a finite set of generators of a bigger group G is called the relative growth of H in G and denoted by G G gH .
    [Show full text]
  • Zeta Functions of Groups and Rings – Recent Developments
    ZETA FUNCTIONS OF GROUPS AND RINGS { RECENT DEVELOPMENTS CHRISTOPHER VOLL Abstract. I survey some recent developments in the theory of zeta functions asso- ciated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations. 1. About these notes Over the last few decades, zeta functions have become important tools in various areas of asymptotic group and ring theory. With the first papers on zeta functions of groups published barely 25 years ago, the subject area is still comparatively young. Recent developments have led to a wealth of results and gave rise to new perspectives on central questions in the field. The aim of these notes is to introduce the nonspecialist reader informally to some of these developments. I concentrate on two types of zeta functions: firstly, zeta functions in subgroup and sub- ring growth of infinite groups and rings, enumerating finite-index subobjects. Secondly, representation zeta functions in representation growth of infinite groups, enumerating finite-dimensional irreducible complex representations. I focus on common features of these zeta functions, such as Euler factorizations, local functional equations, and their behaviour under base extension. Subgroup growth of groups is a relatively mature subject area, and the existing liter- ature reflects this: zeta functions of groups feature in the authoritative 2003 monograph [39] on \Subgroup Growth", are the subject of the Groups St Andrews 2001 survey [16] and the report [18] to the ICM 2006. The book [21] contains, in particular, a substantial list of explicit examples. Some more recent developments are surveyed in [32, Chapter 3].
    [Show full text]
  • Analytic Properties of Zeta Functions and Subgroup Growth
    Annals of Mathematics, 152 (2000), 793–833 Analytic properties of zeta functions and subgroup growth By Marcus du Sautoy and Fritz Grunewald 1. Introduction It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much before, the recent activity grew out of a paper by Dan Segal in [36]. The story so far has been well-documented in Lubotzky’s subsequent survey paper in [30]. In [24] the second author of this article, Segal and Smith introduced the zeta function of a group as a tool for understanding this growth of subgroups. Let an(G) be the number of subgroups of index n in the finitely generated group G and s (G)= a (G)+ + a (G) be the number of subgroups of index N N 1 · · · N or less. The zeta function is defined as the Dirichlet series with coefficients an(G) and has a natural interpretation as a noncommutative generalization of the Dedekind zeta function of a number field: ∞ −s (1.1) ζG(s) = an(G)n nX=1 = G : H −s. | | HX≤G For example, without such a tool it would be difficult to prove that the number of subgroups in the rank-two free abelian group G = Z2 grows as follows: arXiv:math/0011267v1 [math.GR] 1 Nov 2000 s (Z2) π2/12 N 2 N ∼ as N tends to infinity. (Here f(n) g(n) means f(n)/g(n) tends to 1 as n ∼ tends to infinity.) This is a consequence of the expression for the zeta function of the free abelian group of rank d: ζ d (s)= ζ(s) ζ(s d + 1) Z · · · − where ζ(s) is the Riemann zeta function.
    [Show full text]
  • Zeta Functions of Classical Groups and Class Two Nilpotent Groups
    City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 9-2019 Zeta Functions of Classical Groups and Class Two Nilpotent Groups Fikreab Solomon Admasu The Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/3364 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Zeta Functions of Classical Groups and Class Two Nilpotent Groups by Fikreab Solomon A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York. 2019 ii c 2019 Fikreab Solomon All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Mathematics in satisfaction of the dissertation requirements for the degree of Doctor of Philosophy. Professor Gautam Chinta Date Chair of Examining Committee Professor Ara Basmajian Date Executive Officer Professor Abhijit Champanerkar Professor Alexander Gamburd Supervisory Committee THE CITY UNIVERSITY OF NEW YORK iv Abstract Zeta Functions of Classical Groups and Class Two Nilpotent Groups by Fikreab Solomon Advisor: Professor Gautam Chinta This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.
    [Show full text]
  • Combinatorial and Geometric Group Theory
    Combinatorial and Geometric Group Theory Vanderbilt University Nashville, TN, USA May 5–10, 2006 Contents V. A. Artamonov . 1 Goulnara N. Arzhantseva . 1 Varujan Atabekian . 2 Yuri Bahturin . 2 Angela Barnhill . 2 Gilbert Baumslag . 3 Jason Behrstock . 3 Igor Belegradek . 3 Collin Bleak . 4 Alexander Borisov . 4 Lewis Bowen . 5 Nikolay Brodskiy . 5 Kai-Uwe Bux . 5 Ruth Charney . 6 Yves de Cornulier . 7 Maciej Czarnecki . 7 Peter John Davidson . 7 Karel Dekimpe . 8 Galina Deryabina . 8 Volker Diekert . 9 Alexander Dranishnikov . 9 Mikhail Ershov . 9 Daniel Farley . 10 Alexander Fel’shtyn . 10 Stefan Forcey . 11 Max Forester . 11 Koji Fujiwara . 12 Rostislav Grigorchuk . 12 Victor Guba . 12 Dan Guralnik . 13 Jose Higes . 13 Sergei Ivanov . 14 Arye Juhasz . 14 Michael Kapovich . 14 Ilya Kazachkov . 15 i Olga Kharlampovich . 15 Anton Klyachko . 15 Alexei Krasilnikov . 16 Leonid Kurdachenko . 16 Yuri Kuzmin . 17 Namhee Kwon . 17 Yuriy Leonov . 18 Rena Levitt . 19 Artem Lopatin . 19 Alex Lubotzky . 19 Alex Lubotzky . 20 Olga Macedonska . 20 Sergey Maksymenko . 20 Keivan Mallahi-Karai . 21 Jason Manning . 21 Luda Markus-Epstein . 21 John Meakin . 22 Alexei Miasnikov . 22 Michael Mihalik . 22 Vahagn H. Mikaelian . 23 Ashot Minasyan . 23 Igor Mineyev . 24 Atish Mitra . 24 Nicolas Monod . 24 Alexey Muranov . 25 Bernhard M¨uhlherr . 25 Volodymyr Nekrashevych . 25 Graham Niblo . 26 Alexander Olshanskii . 26 Denis Osin . 27 Panos Papasoglu . 27 Alexandra Pettet . 27 Boris Plotkin . 28 Eugene Plotkin . 28 John Ratcliffe . 29 Vladimir Remeslennikov . 29 Tim Riley . 29 Nikolay Romanovskiy . 30 Lucas Sabalka . 30 Mark Sapir . 31 Paul E. Schupp . 31 Denis Serbin . 32 Lev Shneerson .
    [Show full text]
  • Arxiv:1111.0512V4 [Math.GR] 13 May 2013
    MILNOR'S PROBLEM ON THE GROWTH OF GROUPS AND ITS CONSEQUENCES ROSTISLAV GRIGORCHUK Dedicated to John Milnor on the occasion of his 80th birthday. Abstract. 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. 1. Introduction The notion of the growth of a finitely generated group was introduced by A.S. Schwarz (also spelled Schvarts and Svarc))ˇ [S55ˇ ] and independently by Milnor [Mil68b, Mil68a]. Particular studies of group growth and their use in various situations have appeared in the works of Krause [Kra53], Adelson-Velskii and Shreider [AVS57ˇ ], Dixmier [Dix60], Dye [Dye59, Dye63], Arnold and Krylov [AK63], Kirillov [Kir67], Avez [Ave70], Guivarc'h [Gui70, Gui71, Gui73], Hartley, Margulis, Tempelman and other researchers. The note of Schwarz did not attract a lot of attention in the mathematical community, and was essentially unknown to mathematicians both in the USSR and the West (the same happened with papers of Adelson-Velskii, Dixmier and of some other mathematicians). By contrast, the note of Milnor [Mil68a], and especially the problem raised by him in [Mil68b], initiated a lot of activity and opened new directions in group theory and areas of its applications. The motivation for Schwarz and Milnor's studies on the growth of groups were of geometric character.
    [Show full text]
  • Representation Growth Vs Subgroup Growth
    Representation Growth vs Subgroup Growth David A. Craven 18th February 2009 The subgroup growth of finitely generated groups was seen last term, in a lecture of Dan Segal. This time, we see representation growth, and how it is similar to, and different from, subgroup growth. 1 Introduction Let G be a group. If H is a subgroup of G, then let HG denote the core of H in G; this T g is the largest normal subgroup of G contained in H, and can be seen to be g2G H . If H has index n in G, then HG has index at most n!, and in particular, every subgroup of finite index contains a normal subgroup of finite index. We say that G is residually finite if the intersection of all of the (normal) subgroups of finite index is trivial. This is equivalent to saying that for every element x 6= 1, there is some normal subgroup N of finite index such that x2 = N, so that the image of x in G=N is non-trivial. If one wants to study a group via its finite images, then G has to be residually finite, since nothing can be said about the finite residual { the intersection of the subgroups of finite index. Therefore one should study residually finite groups. The example of the Cartesian prod- 1 uct (C2) { a residually finite group with infinitely many subgroups of index 2 { suggests that non-finitely generated groups could be bad. Since the free group on n generators has only finitely many subgroups of index m { a result of Marshall Hall Jr { we see that any finitely generated group has only finitely many subgroups of index n.
    [Show full text]
  • Branch Groups, Orbit Growth, and Subgroup Growth Types for Pro-P
    Forum of Mathematics, Pi (2020), Vol. 8, e10, 26 pages 1 doi:10.1017/fmp.2020.8 BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO- p GROUPS YIFTACH BARNEA1 and JAN-CHRISTOPH SCHLAGE-PUCHTA2 1 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK; email: [email protected] 2 Institut fur¨ Mathematik, Ulmenstr. 69, 18051 Rostock, Germany; email: [email protected] Received 26 February 2019; accepted 9 April 2020 Abstract In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro-p group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between n.log n/2 and en . Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro-p branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type nlog n . 2010 Mathematics Subject Classification: 20E07 (primary); 20E08, 20E18 (secondary) 1. Introduction and results For the rest of this paper, p is a fixed prime and log n D logp n. For a group G, we denote by sn.G/ the number of subgroups of G of index at most n, where if G is a topological group, we mean closed subgroups. The subgroup growth of G is the asymptotic behaviour of sn.G/. See Lubotzky and Segal [8] for the history and background of the study of subgroup growth.
    [Show full text]