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We study the K\"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [ACGT04,ACGT11], that any stage n Bott manifold $M_n$ admits an extremal K\"ahler metric. We also give necessary conditions for $M_n$ to admit a constant scalar curvature K\"ahler metric. We obtain more precise results for stage 3 Bott manifolds, including in particular some interesting relations with c-projective geometry and some explicit examples of almost K\"ahler structures. To place these results in context, we review and develop the topology, complex geometry and symplectic geometry of Bott manifolds. In particular, we study the K\"ahler cone, the automorphism group and the Fano condition. We also relate the number of conjugacy classes of maximal tori in the symplectomorphism group to the number of biholomorphism classes compatible with the symplectic structure., Comment: to appear in Advances in Mathematics

We develop the local-global theory of blocks for profinite groups. Given a field k of characteristic p and a profinite group G, one may express the completed group algebra k [ [ G ] ] as a product ∏ i ∈ I B i of closed indecomposable algebras, called the blocks of G. To each block B of G we associate a pro-p subgroup of G, called the defect group of B, unique up to conjugacy in G. We give several characterizations of the defect group in analogy with defect groups of blocks of finite groups. Our main theorem is a version of Brauer's first main theorem: a correspondence between the blocks of G with defect group D and the blocks of the normalizer N G ( D ) with defect group D.

We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II$_1$ factor with at least one Cartan subalgebra., v3: Final version. Theorem B generalized to all countable groups, exposition improved. To appear in Advances in Mathematics. v2: Corollary G on McDuff factors added

Let G be a finite group and Irr ( G ) the set of irreducible complex characters of G. Let e p ( G ) be the largest integer such that p e p ( G ) divides χ ( 1 ) for some χ ∈ Irr ( G ) . We show that | G : F ( G ) | p ≤ p k e p ( G ) for a constant k. This settles a conjecture of A. Moreto [13, Conjecture 4] . We also study the related problems of the p-parts of conjugacy class sizes of finite groups.

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

In this paper, we identify the moduli space for germs of generic unfoldings of nonresonant linear differential systems with an irregular singularity of Poincar\'e rank $k$ at the origin, under analytic equivalence. The modulus of a given family was determined in \cite{HLR}: it comprises a formal part depending analytically on the parameters, and an analytic part given by unfoldings of the Stokes matrices. These unfoldings are given on "Douady-Sentenac" (DS) domains in the parameter space covering the generic values of the parameters corresponding to Fuchsian singular points. Here we identify exactly which moduli can be realized. A necessary condition on the analytic part, called compatibility condition, is saying that the unfoldings define the same monodromy group (up to conjugacy) for the different presentations of the modulus on the intersections of DS domains. With the additional requirement that the corresponding cocycle is trivial and good limit behavior at some boundary points of the DS domains, this condition becomes sufficient. In particular we show that any modulus can be realized by a $k$-parameter family of systems of rational linear differential equations over $\mathbb C\mathbb P^1$ with $k+1$, $k+2$ or $k+3$ singular points (with multiplicities). Under the generic condition of irreducibility, there are precisely $k+2$ singular points which are Fuchsian as soon as simple. This in turn implies that any unfolding of an irregular singularity of Poincar\'e rank $k$ is analytically equivalent to a rational system of the form $y'=\frac{A(x)}{p_\epsilon(x)}\cdot y$, with $A(x)$ polynomial of degree at most $k$ and $p_\epsilon(x)$ is the generic unfolding of the polynomial $x^{k+1}$., Comment: 45 pages, 19 figures

The C 1 Hartman Theorem states that every C 1 , 1 contraction in R d admits C 1 linearization. In this paper we prove C 1 , β linearizability for C 1 , α contractive (or expansive) random diffeomorphisms with appropriate constants 0 α 1 and 0 β 1 , which implies a random version of the C 1 Hartman Theorem. Moreover, the conjugacy is proved to be tempered. This result strengthens related results even in the deterministic case, lowering the C 1 , 1 condition to C 1 , α and enhancing the C 1 smoothness of the conjugacy to C 1 , β in R d . In the proof we use a smooth weak-stable invariant manifold in the random sense to construct the conjugacy and to overcome difficulties from nonuniformity and measurability.

Aschbacher–Guralnick conjecture states that the number of conjugacy classes of maximal subgroups of a finite group is bounded by the number of conjugacy classes of the group. Attempting to generalize this to the more general framework of subfactors, we are led to investigate the actions of two finite symmetry groups of a subfactor on the first non-trivial relative commutant of the subfactor which are compatible with their actions on the intermediate subfactors. Our first main result is that the actions of these two groups commute, and thus we can formulate a sensible subfactor generalization of Aschbacher–Guralnick conjecture which reduces to the original Aschbacher–Guralnick conjecture in the group subfactor case. In the case of group-subgroup subfactors, our conjecture can be stated in group theory terms as a relative version of Aschbacher–Guralnick conjecture, and we prove that this conjecture is true for solvable groups.

Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass–Jacobi forms of weight one and index two. We prove the convergence of some Maass–Jacobi Poincare series of weight one, and then use these to characterize the semi-holomorphic Maass–Jacobi forms arising from the largest Mathieu group.

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric.

We will study a faithful product type action of G q , the q-deformation of a connected semisimple compact Lie group G, and prove that such an action is induced from a minimal action of the maximal torus T of G q . This enables us to classify product type actions of SU q ( 2 ) up to conjugacy. We also compute the intrinsic group of G q , Ω , the 2-cocycle deformation of G q that is naturally associated with the quantum flag manifold L ∞ ( T \ G q ) .

We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such that $H$ has only 2 conjugacy classes while the conjugacy growth of $G$ is exponential. In particular, conjugacy growth is not a quasi-isometry invariant., The published version of this paper contained an inaccuracy in the proof of Corollary 5.6, which was later corrected in Corrigendum to "Conjugacy growth of finitely generated groups", Adv. Math. 294 (2016), 857-859. This version incorporates all necessary corrections

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order p m with at most c ⋅ m conjugacy classes. This answers a question of L. Pyber.

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g , or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su ( m ) and products of conjugacy classes in SU ( m ) contain an open set, in terms of the highest multiplicities of eigenvalues. For su ( m ) and SU ( m ) we show L 2 -singular dichotomy: The convolution of invariant measures on adjoint orbits, or conjugacy classes, is either singular to Haar measure or in L 2 .

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L 2 -singular dichotomy: Either μ h k ∈ L 2 ( G ) or μ h k is singular to the Haar measure on G. The minimum exponent k for which μ h k ∈ L 2 is specified; it depends on Lie properties of the element h ∈ G . As a corollary, we complete the solution to a classical problem – to determine the minimum exponent k such that μ k ∈ L 1 ( G ) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.

We give a complete classification up to cocycle conjugacy of uniformly outer actions of on UHF algebras. In particular, it is shown that any two uniformly outer actions of on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy

We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group $G(e,e,r)$. For the particular case $e=2$ (resp. $r=2$), our lattice coincides with the lattice of simple elements for the type $D_n$ (resp. $I_2(e)$) dual braid monoid. Using this lattice, we construct a Garside structure for the braid group $B(e,e,r)$. As a corollary, one may solve the word and conjugacy problems in this group., 38 pages, 15 figures; second version with extended introduction

We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M.P. Holland and the first author. We show that these algebras provide a natural setting for the ‘middle convolution’ operation introduced by N.M. Katz in his book ‘Rigid local systems’, and put in an algebraic setting by M. Dettweiler and S. Reiter, and H. Volklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne–Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation A 1 A 2 … A k = 1 with the A i in prescribed conjugacy classes in GL n ( C ) .

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate $x^{c\log x}$. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers., Comment: 27 pages, a small change in title, final revision, to appear in Adv. Math

Let S(X, Y) be the germ of an analytic function of two complex variables near the origin. To decide whether f is irreducible or not, we can apply a succession of blowing-ups to desingularize f and see if the proper transform off ever bifurcates, and then use the fact that f is irreducible iff such a bifurcation never takes place. Alternatively, we can use Newton’s theorem (also called Puiseux’s theorem) to completely factor S into linear factors in Y by allowing fractional power series in X and then collate these factors into conjugacy classes, and now use the fact that fis irreducible iff there is only one conjugacy class. In a recent conversation, T. C. Kuo of the University of Sydney in Australia asked me the following question

This paper begins with a survey of the known results about root systems. There follows the definition of hyperbolic root systems, and the explicit description of the imaginary roots of such a system as the points of intersection of a lattice and a cone. Finally there is a proof of the conjugacy of bases in a symmetrizable root system by the extended Weyl group.

The existence of solutions is established for a very general class of problems in the calculus of variations and optimal control involving ordinary differential equations or contingent equations. The theorems, while relatively simple to state, cover, besides the more classical cases, problems with considerably weaker assumptions of continuity or boundedness. For example, the cost functional may only be lower semicontinuous in the control and may approach + ∞ as one nears certain boundary points of the control region; both endpoints in the problem may be “free”. Earlier results of Cesari, Olech and the author are thereby extended. The development is based on the theory of convex integral functionals and their conjugates. The first step is to show that, for purposes of existence theory, the problem can be reduced to a simpler model where control variables are not present as such. This model, resembling a classical problem of Bolza in the calculus of variations, but where the functions are extended-real-valued, is then investigated using, above all, the conjugacy correspondence between generalized Lagrangians and Hamiltonians.

Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the given conjugacy class. Our main result is to prove that m is bounded above by n-2 in the situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the first. As a corollary, we show that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, as defined by Charney, and so we also obtain a polynomial-time algorithm for computing this geodesic length., 15 pages. Journal

We consider the canonical map from the Calogero-Moser space to symmetric powers of the affine line, sending conjugacy classes of pairs of n by n matrices to their eigenvalues. We show that the character of a natural C^*-action on the scheme-theoretic zero fiber of this map is given by Kostka polynomials., Comment: 12pp., LaTeX

Let G be a finite solvable group, and let F ( G ) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G / F ( G ) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r ′ -parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes.

This paper is part of a series [S, 131 devoted to the study of a class G of irreducible perverse sheaves (called character sheaves) on a connected reductive algebraic group G. (The numbering of chapters, sections, and references will continue that of [S, 131.) This paper is a step towards the classification of character sheaves on G. One of the main results is the following one: under certain assumptions, there is a natural surjective map with finite fibers from G to the set of all pairs (9, c) (up to conjugacy by the Weyl group), where 9 is a tame local system on the maximal torus and c is a “two-sided cell’ in the stabilizer W:’ of 9 in the Weyl group. The assumptions made on G are

1.1 Let G(F,) be the group of Fq rational points of a connected reductive algebraic group G defined over a finite field F,. Throughout this paper we shall assume that the characteristic of the ground field is large. The following is part of Problem II in [L9]. “Let p be an irreducible complex character of G(F,). Show that there is a unique umpotent class c’ in G which has the property that Cga c’CFqI p(g) # 0 and has maximal dimension among classes with this property.” One of the results of this paper is a solution of this problem; we also prove the following refinement. If g E GF is such that Tr(g, p) # 0 then the unipotent part of g lies in C or in a conjugacy class of dimension

We construct a generator system of the annihilator of the generalized Verma module of g l ( n , C ) induced from any character of any parabolic subalgebra as an analogue of minors and elementary divisors. The generator system has a quantization parameter e and it generates the defining ideal of the conjugacy class of square matrices at the classical limit e = 0 .

We study the structure constants of the class algebra RZ(Γn) of the wreath products Γn associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on RZ(Γn) gives rise to the graded ring GΓ(n) with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat–Higman ring GΓ. The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of SL2(C). The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5., Comment: 36 pages

Let p ⩾ 2 be a prime number and let Z p be the ring of all p-adic integers. For all α , β , z ∈ Z p , define T α , β ( z ) = α z + β . It is shown that the dynamical system ( Z p , T α , β ) is minimal if and only if α ∈ 1 + p r p Z p and β is a unit, here r p = 1 (respectively r p = 2 ) if p ⩾ 3 (respectively if p = 2 ), and that when it is minimal, it is strictly ergodic and topologically conjugate to ( Z p , T 1 , 1 ) with an analytic and isometric conjugacy. More importantly, when the system is not minimal, we find all its strictly ergodic components. As application, monomial systems S n , ρ ( z ) = ρ z n on the group 1 + p Z p are also discussed.

A new presentation of the n -string braid group B n is studied. Using it, a new solution to the word problem in B n is obtained which retains most of the desirable features of the Garside–Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.