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Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

When we think about symmetries, regular geometrical objects are the ones which come first to mind. While it is true that the concept of symmetry has a much wider meaning in mathematics and physics, the importance of symmetries of basic geometrical objects should not be taken lightly. They play pivotal role in the study of many kinds of systems, from molecules to crystals and beyond. In this chapter, we discuss the symmetry groups associated with some regular geometrical object in two and three dimensions, and the representations of those groups. SYMMETRIES OF REGULAR POLYGONS We have mentioned before that the symmetry group of a regular polygon with n vertices is called D n , and that it contains 2 n elements, among which there is of course the identity element, n − 1 non-trivial rotations and n reflections. In Section 8.9.5, we discussed the conjugacy class structure of the D n groups. Taking the cue from there, we discuss here the irreducible representations of D n groups. As is clear from the discussion of Section 8.9.5, this discussion will have to be different for odd and even values of n . Representations of Dn for odd n We take n = 2 r +1, where r is an integer. THEOREM 10.1 The group Dn, for n = 2 r + 1 , admits two 1-dimensional and r 2-dimensional irreps. PROOF: We saw in Section 8.9.5 that the number of conjugacy classes is r + 2. The total number of elements is 2 n = 4 r +2. There is only one way that the equation for the dimensionalities of irreps, Eq. (9.86) can be satisfied, and the solution is given in the statement of this result. The characters for the 1-dimensional irreps are obvious. One of these is the trivial representation for which all characters are equal to 1. The other has χ = +1 for the even permutations and χ = −1 for the odd permutations. Before trying to give the characters of the 2-dimensional irreps for general D n , let us try to find the character table for D 5 as an example.

A finite group can be generated in many different ways. In this paper we consider a few methods of generating finite simple groups and in particular we focus on those of interest to the authors especially the second author and his research group. These methods are concerned with ranks of conjugacy classes of elements, ( p, q, r )-, nX- complementary generation and exact spread of finite non-abelian simple groups. We also give some examples of results that were established by the authors on generation of some finite non-abelian simple groups.

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group G. They can be regarded both as a generalization of normal subgroups as well as a generalization of lattices. As such, it is intriguing to extend results from the theories of normal subgroups and of lattices to the context of IRS. Another approach is to analyse and then use the space IRS(G) as a compact G-space in order to establish new results about lattices. The second approach has been taken in the work [7s12], that came to be known as the seven samurai paper. In these lecture notes we shall try to give a taste of both approaches.

Introduction Alfred Tarski in 1940 made three well-known conjectures concerning nonabelian free groups (see Section 2). There had been various partial solutions until complete positive solutions were presented during the past 15 years by Kharlampovich and Myasnikov (see [51]–[59]) and independently by Z. Sela (see [78]–[83]). In the Kharlampovich- Myasnikov approach the proof arose from a detailed study of fully residually free groups (called limit groups in Sela's approach), the development of algebraic geometry over free groups, and an elimination process involving solutions of equations over free groups based on work of Makhanin and Razborov (see [51]–[59]). These steps were mirrored, with somewhat different terminology, by Sela, who called his approach diophantine geometry over free groups. The positive solution of the Tarski conjectures provides a straightforward proof of Magnus's theorem in surface groups which we present. This result was proved directly by J. Howie [46] and independently by O. Bogopolski [10]. We will present this proof in Section 4. This type of proof leads to several different types of questions. • Which additional nontrivial free group results are true in surface groups but difficult to obtain directly? • What first-order properties of nonabelian free groups are true beyond the class of elementary free groups? After showing a proof of Magnus's Theorem based on the solution of the Tarski problems we give several examples of other free group results holding in surface groups. Using this technique we give a proof of a theorem of D. Lee on C-test words. We then consider and prove certain other results that hold in elementary free groups, in particular surface groups, including the retract theorem of Turner [86] and the property of conjugacy separability. After this we turn to the second type of question and survey a large number of recent results. In particular we first consider groups satisfying certain quadratic properties that we call Lyndon properties and show that the class of groups satisfying these properties are closed under many amalgam constructions.

The aim of this chapter is to explain how one can obtain information regarding the membership of a classical weight one eigenform in a Hida family from the geometry of the Eigencurve at the corresponding point. We show, in passing, that all classical members of a Hida family, including those of weight one, share the same local type at all primes dividing the level. 1. Introduction Classical weight one eigenforms occupy a special place in the correspondence between Automorphic Forms and Galois Representations since they yield two dimensional Artin representations with odd determinant. The construction of those representations by Deligne and Serre [5] uses congruences with modular forms of higher weight. The systematic study of congruences between modular forms has culminated in the construction of the p -adic Eigencurve by Coleman and Mazur [4]. A p -stabilized classical weight one eigenform corresponds then to a point on the ordinary component of the Eigencurve, which is closely related to Hida theory. An important result of Hida [11] states that an ordinary cuspform of weight at least two is a specialization of a unique, up to Galois conjugacy, primitive Hida family. Geometrically this translates into the smoothness of the Eigencurve at that point (in fact, Hida proves more, namely that the map to the weight space is etale at that point). Whereas Hida's result continues to hold at all non-critical classical points of weight two or more [13], there are examples where this fails in weight one [6].

In this chapter we describe what quasiconformal surgery is about and explain the surgeries which were developed as the first occurrences of this technique: the parametrization of hyperbolic components of the Mandelbrot set and the proof of the No Wandering Domains Theorem. Generalities What is known as quasiconformal surgery in holomorphic dynamics is a technique commonly used to construct holomorphic maps with prescribed dynamics. The ‘prescribed dynamics’ are given by a map f which in general is not holomorphic, although it may be. We shall refer to f as the model map . The word surgery appears because one may need to ‘cut’ and ‘paste’ different spaces and maps together to construct f . This is usually the first step in the construction and is known as topological surgery . We are leaving the holomorphic world in order to have a greater choice for our models, and then checking whether the model map has a ‘holomorphic dynamical copy’, i.e. whether there exists a holomorphic map conjugate to f . The main tool for obtaining ‘holomorphic dynamical copies’ is to apply the Integrability Theorem (Theorem 1.28), which provides a quasiconformal conjugacy to return to the holomorphic setting (see the Key Lemma (Lemma 1.39)). It follows that we should look for models in the space of quasiregular maps (see Proposition 1.37).

We now consider properties like generation, conjugacy, classification, connectedness and dimension of centralizers in connected reductive groups. It turns out that the situation is easiest for semisimple elements. Semisimple elements Recall from Corollary 6.11(a) that every semisimple element of a connected group lies in some maximal torus. More precisely we have: Proposition 14.1 Let G be connected, s ∈ G semisimple, T ≤ G a maximal torus. Then s ∈ T if and only if T ≤ C G (s)°. In particular, s ∈ C G (s)° . Proof As T is abelian, s ∈ T if and only if T ≤ C G (s) , which is equivalent to T ≤ C G (s) ° as T is connected. We remark that in contrast, for u ∈ G unipotent, u may not be in C G (u) °. See Exercise 20.10 for an example in Sp 4 over a field of characteristic 2. We now determine the structure of centralizers of semisimple elements: Theorem 14.2 Let G be connected reductive, s ∈ G semisimple, T ≤ G a maximal torus with corresponding root system Φ. Let s ∈ T and Ψ ≔ {α ∈ Φ | α(s) = 1}. Then: (a) C G (s) ° = 〈 T,U α ; | α ∈ Ψ〉. (b) C G (s) = 〈 T,U α ,ẇ | α ∈ Ψ, w ∈ W with s w = s 〉. Moreover, C G (s)° is reductive with root system Ψ and Weyl group W 1 = 〈s α | α ∈ Ψ〉 .

A conjugacy class D of 3-transpositions in the group G is a class of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. Such groups were introduced and studied by Bernd Fischer who classified all finite 3-transposition groups with no nontrivial normal, solvable subgroups. His work was of great importance in the classification of finite simple groups.The basic example of a class of 3-transpositions is the class of transpositions in any symmetric group. This was the only class which Fischer originally considered, but Roger Carter pointed out that examples could be found in several of the classical groups as well. The transvections of symplectic groups over GF(2) form a class of 3-transpositions, so additionally any subgroup of the symplectic group generated by a class of transvections is also a 3-transposition group. The symmetric groups arise in this way as do the orthogonal groups over GF(2). Symplectic transvections over GF(2) are special cases of unitary transvections over GF(4), and this unitary class is still a class of 3-transpositions. The final classical examples are given by the reflection classes of orthogonal groups over GF(3).

We survey recent computational results concerning the Monster sporadic simple group. The main results are: progress towards a complete classification of the maximal subgroups, including showing that L 2 (27) is not a subgroup; showing that the 196882-dimensional module over GF (2) supports a quadratic form; a complete set of explicit conjugacy class representatives; small representations of most of the maximal subgroups; and a partial classification of the ‘nets’ (in the sense of Norton). Introduction Our aim in this paper is to update the survey by describing the various explicit computations which have been performed in the Monster group, and the new information about the Monster which has resulted from these calculations. We begin by summarising for the benefit of readers who do not have that paper to hand. The smallest matrix representations of the Monster have dimension 196882 in characteristics 2 and 3, and dimension 196883 in all other characteristics. Three of these representations (over the fields of orders 2, 3, and 7) are now available explicitly. It is hoped that the data and programs to manipulate them will be made available in the next release of M agma . The generating matrices are stored in a compact way, and never written out in full. The basic operation of the system is to calculate the action of a generator on a vector of the underlying module. Our first construction was carried out over the field GF (2) of two elements in the interests of speed, and proceeded by amalgamating various 3-local subgroups.

The second of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains a series of expository essays and research papers around the subject matter of a Newton Institute Semester on Model Theory and Applications to Algebra and Analysis. The articles convey outstanding new research on topics such as model theory and conjectures around Mordell-Lang; arithmetic of differential equations, and Galois theory of difference equations; model theory and complex analytic geometry; o-minimality; model theory and non-commutative geometry; definable groups of finite dimension; Hilbert's tenth problem; and Hrushovski constructions. With contributions from so many leaders in the field, this book will undoubtedly appeal to all mathematicians with an interest in model theory and its applications, from graduate students to senior researchers and from beginners to experts.

In this final chapter, we will describe the use of the large sieve to study the average distribution of (geometric) Frobenius conjugacy classes in Galois groups of coverings of algebraic varieties over finite fields, or equivalently in a more geometric language that we will use instead, in finite monodromy groups of sheaves obtained by reduction of integral l-adic sheaves. This sieve is a good example (in fact, the most interesting at the moment) of a coset (conjugacy) sieve, as defined in Section 3.3. This type of sieve was introduced in [ 80 ], and its strengthening was the motivation for the paper from which this book evolved. We will recall enough of the previous work to make the argument independent of results in [ 80 ]. As explained in Example 4.10, there is nothing to prevent adapting the ideas to sieve for Frobenius conjugacy classes over number fields, except that really good results depend at present on assuming some form of the Generalized Riemann Hypothesis (though weaker unconditional bounds are possible, see D. Zywina's preprint ‘The large sieve and Galois representations’, 2007). Contrary to what we have done in all previous applications of the sieve, we have not attempted to give entirely self-contained definitions ; here, we need to introduce some ‘black boxes’. Hopefully, the examples of applications (which we can, and do, describe from scratch) will be sufficiently interesting to encourage interested readers to get better acquainted with the foundations and in particular with Deligne's work on the Riemann Hypothesis over finite fields.

Summary We survey conjugacy results in groups of finite Morley rank, mixing unipotence, Carter, and Sylow theories in this context. Introduction When considering certain classes of groups one might expect conjugacy theorems, and the class of groups of finite Morley rank is not an exception to this. The study of groups of finite Morley rank is mostly motivated by the Algebricity Conjecture, formulated by G. Cherlin and B. Zilber in the late seventies, which postulates that infinite simple groups of this category are isomorphic to algebraic groups over algebraically closed fields. The model-theoretic rank involved appeared in the sixties when M. Morley proved his famous theorem on the categoricity in any uncountable cardinal of first order theories categorical in one uncountable cardinal [Mor65]. He introduced for that purpose an ordinal valued rank, later shown to be finite by J. Baldwin in the uncountably categorical context [Bal73], and this rank can be seen as an abstract version of the Zariski dimension in algebraic geometry over an algebraically closed field. In particular, the category of groups of finite Morley rank encapsulates finite groups and algebraic groups over algebraically closed fields. One of the most basic tools for analyzing finite groups is Sylow theory, and in algebraic groups semisimplicity and unipotence theory play a similar role. It is thus not surprising to see these two theories, together with all conjugacy results they sugest, having enormous and close developments in the more abstract category of groups of finite Morley rank.

In this book, which arose from an MSRI research workshop cosponspored by the Clay Mathematical Institute, leading experts give an overview of several areas of dynamical systems that have recently experienced substantial progress. In symplectic geometry, a fast-growing field having its roots in classical mechanics, Cieliebak, Hofer, Latschev and Schlenk give a definitive survey of quantitative techniques and symplectic capacities, which have become a central research tool. Fisher's survey on local rigidity of group actions is a broad and up-to-date account of a flourishing subject built on the fact that for actions of noncyclic groups, topological conjugacy commonly implies smooth conjugacy. Other articles by Eigen, Feres, Kochergin, Krieger, Navarro, Pinto, Prasad, Rand and Robinson cover subjects in hyperbolic, parabolic and symbolic dynamics as well as ergodic theory. Among the specific areas of interest are random walks and billiards, diffeomorphisms and flows on surfaces, amenability and tilings. The articles are complemented by a fifty-page commented problem list, compiled by the editor with the help of numerous specialists. Several sections of this list focus on problems beyond the areas covered in the surveys, and all are sure to inspire and guide further research.

Let G = SL(2, Qp). Let k > 2 and consider the space Hom(Fk, G) where Fk is the free group on k generators. This space can be thought of as the space of all marked k-generated subgroups of G, i.e., subgroups with a given set of k generators. There is a natural action of the group Aut(Fk) on Hom(Fk, G) by pre-composition. I will prove that this action is ergodic on the subset of dense subgroups. This means that every measurable property either holds or fails to hold for almost all k-generated subgroups of G together. Speaker: Volodymyr Nekrashevych (Texas A&M) Title: Self-similar groups, limit spaces and tilings Abstract: We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting “nicely” on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilings, on leaves of the solenoid associated with the covering. Speaker: Olga Kharlampovich (McGill) Title: Undecidability of Markov Properties Abstract: A group-theoretic property P is said to be a Markov property if it is preserved under isomorphism and if it satisfies: 1. There is a finitely presented group which has property P . 2. There is a finitely presented group which cannot be embedded in any finitely presented group with property P . Adyan and Rabin showed that any Markov property cannot be decided from a finite presentation. We give a survey of how this is proved. Speaker: Alexei Miasnikov (McGill) Title: The conjugacy problem for the Grigorchuk group has polynomial time complexity Abstract: We discuss algorithmic complexity of the conjugacy problem in the original Grigorchuk group. Recently this group was proposed as a possible platform for cryptographic schemes (see [4, 15, 14]), where the algorithmic security of the schemes is based on the computational hardness of certain variations of the word and conjugacy problems. We show that the conjugacy problem in the Grigorchuk group can be solved in polynomial time. To prove it we replace the standard length by a new, weighted length, called the norm, and show that the standard splitting of elements from St(1) has very nice metric properties relative to the norm. Speaker: Mark Sapir (Vanderbilt) Title: Residual finiteness of 1-related groups Abstract: We prove that with probability tending to 1, a 1-relator group with at least 3 generators and the relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory, non-trivial results about Brownian motions, and non-trivial algebraic geometry (and Galois theory). This is a joint work with A. Borisov and I. Kozakova. Speaker: Dmytro Savchuk (Texas A&M) Title: GAP package AutomGrp for computations in self-similar groups and semigroups: functionality, examples and applications Abstract: Self-similar groups and semigroups are very interesting from the computational point of view because computations related to these groups are often cumbersome to be performed by hand. Many algorithms related to these groups were implemented in AutomGrp package developed by the authors (available at http://www.gap-system.org/Packages/automgrp.html). We describe the functionality of the package, give some examples and provide several applications. This is joint with Yevgen Muntyan Speaker: Benjamin Steinberg (Carleton) Title: The Ribes-Zalesskii Product Theorem and rational subsets of groups

In this chapter we present results of [48] proving that, given any p ∈ [0,1]\ℚ there exists a unimodal map ƒ such that (S 1, R ρ ) is a factor of (ω(c, ƒ),ƒ) (recall Remark 3.3.2 for the definition of factor). One might ask whether one can obtain the stronger result of conjugacy? As S 1 is not homeomorphic to a Cantor set, and in this setting ω(c, ƒ) is indeed a Cantor set, a conjugacy is not possible. See [48, 47] for further details and results. Chapter 3 and Sections 6.1, 7.2, and 8.3 contain background material for this chapter. Adding Machines in Unimodal Maps Given a unimodal map ƒ with turning point c and kneading map Q(k), we construct an adding machine (ω, P ) from the sequence of cutting times {S k }. In the event that lim k→∞ Q(k) = ∞, we have that (ω(c, ƒ),ƒ) is a factor of (Ω, P ) (Theorem 11.1.15). The condition lim k→∞ Q(k) = ∞ is not required to define the adding machine (ω, P ), but rather comes into play for the continuity of the map P . In this section we provide only the information on the adding machine (ω, P ) needed to obtain Theorem 11.1.15; see Section 13.3 for a more detailed discussion of (ω, P ). Let S 0 1 2 3 ,… be the sequence of cutting times for some unimodal map ƒ (recall it is always the case that S 0 = 1 and S 1 = 2). We do not assume that lim k→∞ Q(k) = ∞.

Introduction Historical sketch of Kac–Moody theory .— Kac-Moody theory was initiated in 1968, when V. Kac and R. Moody independently defined infinite-dimensional Lie algebras generalizing complex semi-simple Lie algebras. Their definition is based on Serre's presentation theorem describing explicitly the latter (finite-dimensional) Lie algebras [Hu1, 18.3]. A natural question then is to integrate Kac–Moody Lie algebras as Lie groups integrate real Lie algebras, but this time in the infinite-dimensional setting. This difficult problem led to several propositions. In characteristic 0, a satisfactory approach consists in seeing them as subgroups in the automorphisms of the corresponding Lie algebras [KP1,2,3]. Thisway, V. Kac and D. Peterson developed the structure theory of Kac–Moody algebras in complete analogy with the classical theory: intrinsic definition and conjugacy results for Borel (resp. Cartan) subgroups, root decomposition with abstract description of the root system… Another aspect of this work is the construction of generalized Schubert varieties. These algebraic varieties enabled O. Mathieu to get a complete generalization of the character formula in the Kac–Moody framework [Mat1]. To this end, O. Mathieu defined Kac–Moody groups over arbitrary fields in the formalism of ind-schemes [Mat2]. Combinatorial approach . — Although the objects above – Kac–Moody groups and Schubert varieties – can be studied in a nice algebro-geometric context, we will work with groups arising from another, more combinatorial viewpoint. All of this work is due to J. Tits [T4,5,6,7], who of course contributed also to the previous problems.

For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria and ours for one-dimensional expanding dynamics. For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria and ours for one-dimensional expanding dynamics.

Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure.

Introduction Many interesting finite geometries, graphs and designs admit automorphism groups of low rank. In fact, it was a study of the rank 3 case which led to the discoveries and constructions of some of the sporadic simple groups (see [Gor82]). For several classification problems about graphs or designs, the case where the automorphism group is almost simple is of central importance, and many of the examples have a transitive automorphism group of low rank. This is the case, for example, for the classification problems of finite distance-transitive graphs [BCN89, PSY87], and of finite flag-transitive designs [BDD88, BDDKLS90]. This book presents a complete classification, up to conjugacy of the point stabilizers, of the faithful transitive permutation representations of rank at most 5 of the sporadic simple groups and their automorphism groups. These results, summarized in Chapter 5, filled a major gap in the existing classification results for finite, low rank, transitive permutation groups. For each representation classified, we also give the collapsed adjacency matrices (defined in Section 2.3) for all the associated orbital digraphs. We use these collapsed adjacency matrices to classify the vertex-transitive, distance-regular graphs for these low rank representations, and discover some new distance-regular graphs of diameter 2 (but of rank greater than 3) for the O'Nan group O'N , the Conway group Co 2 , and the Fischer group Fi 22 .