Discrete and Profinite Groups Acting on Regular Rooted Trees Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Olivier Siegenthaler aus Lausanne G¨ottingen, den 31. August 2009 Referent: Prof. Dr. Laurent Bartholdi Korreferent: Prof. Dr. Thomas Schick Tag der m¨undlichen Pr¨ufung: den 28. September 2009 Contents Introduction 1 1 Foundations 5 1.1 Definition of Aut X∗ and ...................... 5 A∗ 1.2 ZariskiTopology ............................ 7 1.3 Actions of X∗ .............................. 8 1.4 Self-SimilarityandBranching . 9 1.5 Decompositions and Generators of Aut X∗ and ......... 11 A∗ 1.6 The Permutation Modules k X and k X ............ 13 { } {{ }} 1.7 Subgroups of Aut X∗ .......................... 14 1.8 RegularBranchGroups . 16 1.9 Self-Similarity and Branching Simultaneously . 18 1.10Questions ................................ 19 ∗ 2 The Special Case Autp X 23 2.1 Definition ................................ 23 ∗ 2.2 Subgroups of Autp X ......................... 24 2.3 NiceGeneratingSets. 26 2.4 Uniseriality ............................... 28 2.5 Signature and Maximal Subgroups . 30 2.6 Torsion-FreeGroups . 31 2.7 Some Specific Classes of Automorphisms . 32 2.8 TorsionGroups ............................. 34 3 Wreath Product of Affine Group Schemes 37 3.1 AffineSchemes ............................. 38 3.2 ExponentialObjects . 40 3.3 Some Hopf Algebra Constructions . 43 3.4 Group Schemes Corresponding to Aut X∗ .............. 45 3.5 Iterated Wreath Product of the Frobenius Kernel . 46 i Contents 4 Central Series and Automorphism Towers 49 4.1 Notation................................. 49 4.2 CentralSeries.............................. 51 4.3 Automorphism and Normalizer Towers . 54 5 Hausdorff Dimension 57 5.1 Definition ................................ 57 5.2 Layers .................................. 58 5.3 ComputingDimensions . 60 5.4 Dimension1............................... 62 6 Congruence Problems 65 6.1 StructureoftheRigidKernel . 65 6.2 AnExample............................... 67 7 Twisted Twins 69 7.1 The Closure of the Grigorchuk Group . 69 7.2 Generators and Branchness of G2 ................... 71 7.3 A Presentation of G2 .......................... 74 Appendices 77 A Examples 77 A.1 TheGrigorchukGroup. 77 A.2 The Twisted Twin of the Grigorchuk Group . 78 A.3 TheGupta-SidkiGroup . 79 A.4 The Fabrykowski-Gupta Group . 80 A.5 AnotherBurnsideGroup. 81 A.6 The Binary Adding Machine . 83 A.7 TheBrunner-Sidki-VieiraGroup . 84 A.8 TheHanoiTowerGroup . 85 B Hausdorff Dimension of Some Groups Acting on the Binary Tree 87 B.1 Introduction............................... 87 B.2 HausdorffDimension . 88 B.3 SpinalGroups.............................. 88 B.4 MainTheorem ............................. 89 B.5 Finitely Generated Groups of Irrational Hausdorff Dimension . 90 B.6 Full-Dimensional Finitely Generated Groups . 91 B.7 ProofoftheMainTheorem . 93 ii Contents C The Congruence Subgroup Problem for Branch Groups 99 C.1 Introduction............................... 99 C.2 The Congruence Subgroup Problem . 106 C.3 Examples ................................112 D The Twisted Twin of the Grigorchuk Group 123 D.1 Introduction...............................123 D.2 BasicDefinitionsandNotation . .124 D.3 DefinitionoftheGroup . .125 D.4 APresentation .............................127 D.5 TheCongruenceKernel . .130 D.6 Germs ..................................132 D.7 TheLowerCentralSeries . .136 D.8 TheSchurMultiplier . .138 D.9 GAPComputations . .140 Bibliography 145 Curriculum Vitæ 149 iii Introduction About thirty years ago, people’s attention has been attracted to groups acting on rooted trees. For, some particularly simple and elegant constructions of Burnside groups showed up (due to Aleshin [Ale72], Sushchansky [Suˇs79], and then to Grigorchuk [Gri80] and Gupta and Sidki [GS83a]), as well as the first example of a group of intermediate growth. The “first” Grigorchuk group [Gri83] was the first example, and so far, groups acting on rooted trees remain the only source for such groups. There has been a number of articles since then, with new examples, general- izations, and so on. Nowadays, it is hard to embrace globally all groups acting on rooted trees, because of their diversity. Apart from the groups mentioned above, there are also many free groups [Bha95,AV05], groups with non-uniformly exponential growth [Wil04], groups which are amenable yet not subexponentially amenable [BV05], . Based on the observation that a regular rooted tree contains copies of itself, people considered groups containing (up to finite index) copies of themselves, or groups which almost embed in their own direct powers. From this emerged the definitions of self-similar groups, and of regular branch groups. Both notions express the fact that the group almost replicates itself at each vertex of the tree. Many of the main examples, including Grigorchuk’s and Gupta and Sidki’s, are both self-similar and regular branch. For now, the monograph of Nekrashevych [Nek05] is the only one on the topic, and it is oriented towards the correspondence between contracting groups and dynamical systems. We still lack a good reference including the profinite aspects of groups acting on rooted trees, probably because less is known about it. This work is a step in this direction. Overview The first two chapters form the core of this thesis, where we explain in detail our approach to the topic. The following five chapters are mostly independent from one another, and treat several aspects of groups acting on rooted trees, under the light of the core chapters. The first appendix contains a summary of examples treated throughout. It can be used as a reference, and contains a quick introduction to each group. The three publications to which I contributed, and which complement this work, are reprinted as appendices. 1 The Core We start with the automorphism group Aut X∗ of a regular X -ary rooted tree. | | This is a profinite group when the set X is finite, and we concentrate on that case. We then fix a finite field k and consider the set of all continuous functions A∗ Aut X∗ k. This is naturally an infinite dimensional k-algebra. → This work is based on a description of closed subgroups G of Aut X∗ by means of the ideal (G) of functions in vanishing on G. This reminds of algebraic I A∗ geometry, and we explain the analogy in detail. Also, we consider only continuous functions Aut X∗ k, that is, functions which factor through a finite quotient of → Aut X∗. Therefore, the ideal (G) can be seen as the set of “forbidden patterns” I corresponding to G, as was introduced in [Gri05] (see also [AHKS07,ˇ Appendix]). Identifying the vertices of the tree with the free monoid on X∗ yields an operation on vertices, corresponding to concatenation of words. In turn, this gives operations on the group Aut X∗ and dually on . From this, we define the A∗ crucial notions of branching and self-similarity of groups and of algebras. The group Aut X∗ and the algebra are not finitely generated, but only finitely A∗ many generators are needed when we consider Aut X∗ and as a branching A∗ objects. As a main object of study, we consider the correspondence between closed subsets of Aut X∗ and ideals in . In particular, closed subgroups correspond to A∗ Hopf ideals, and any knowledge about the ideal (G) (the annihilator) associated I to a closed group G Aut X∗ might yields information about G. A situation ≤ which is particularly well suited to this approach arises when G is regular branch. We shall see that in this case (and only in this case), the ideal (G) is finitely I generated as a branching ideal. Therefore G is characterized by a finite number of equations or “patterns” [Gri05]. This agrees with a result of Sunic [Sun07].ˇ In Chapter 2 we consider the following special case. Let X = k = Fp be ∗ the field with p elements, with p prime. We consider the group Autp X of automorphisms whose activity at each vertex is in the additive group of Fp. To a closed subgroup G of Aut X∗ corresponds an ideal (G) in the algebra of p Ip P∗ continuous functions Aut X∗ F . Such ideals behave better than in the case p → p of Aut X∗, in that they always have “nice generating sets”. At the end of the chapter, we give some conditions on (G) for G to be torsion-free, or for the Ip spinal subgroup of G to be torsion. ∗ Many examples of self-similar groups are discrete subgroups of Autp X . In Appendix A we give generators for the annihilator of several groups, including the well-known examples of Grigorchuk and of Gupta and Sidki. Five Directions Chapter 3 is almost independent from the rest of this work, and is the beginning of a possible new development of the topic, relating to algebraic geometry. Modern algebraic geometry contains the whole of commutative algebra. Indeed, to any algebra (commutative, with unit), there is a corresponding affine scheme. Within this correspondence, Hopf algebras are associated to affine group schemes. We briefly recall the relevant definitions, and introduce the wreath product of affine 2 Introduction group schemes. We prove that the constant scheme associated to Aut Xn can be seen as the iterated wreath product of the constant scheme associated to Sym(X). This is satisfactory, and perhaps not very surprising. More interesting is the case of the infinitesimal scheme αp, the “Frobenius kernel”. Although this scheme has only one geometric point on fields, we shall see that the Lie algebra of the iterated wreath product of αp is isomorphic to the Lie algebra obtained from the n lower central series of Autp X . Building on previous work by Kaloujnine [Kal95], Rozhkov [Roz96], Bartholdi and Grigorchuk [BG00,Bar05] and Bartholdi and Sidki [BS06], we investigate in Chapter 4 the ideals associated to the lower central series and automorphism towers of some specific examples. We shall see that in some cases, these series of groups present surprising similarities. In Chapter 5, we define the Hausdorff dimension of subgroups of Aut X∗. I investigated this subject some time ago, providing the first explicit examples of groups of irrational dimension in [Sie08].