We define and study some generalisations of pure 𝔤 -braid groups, for any complex reductive Lie algebra 𝔤 . They naturally occur in the theory of isomonodromic deformations for meromorphic connections with irregular singularities on principal bundles over Riemann surfaces, covering the general untwisted case, going beyond the case of generic irregular types. These generalised braid groups make up local pieces of the wild mapping class groups, which in turn extend the usual mapping class groups and govern the braiding of Stokes data. We establish a general product decomposition for these local wild mapping class groups, and in all classical cases define a fission tree governing the decomposition; in particular in type D we will find a factor which is not isomorphic to any pure braid group coming from a root system. In type A , the fission tree and the pure braid group operad yields a proof of the corresponding “multi-scale” braiding conjecture.
For any unbranched double covering of a compact Riemann surface, we study the associated character varieties that are unitary in the global sense, which we call GL n ⋊ 〈 σ 〉 -character varieties. We introduce k > 0 punctures on the surface, restrict the monodromies around the punctures to generic semi-simple conjugacy classes in GL n , and compute the E-polynomials of these character varieties using the character table of GL n ( q ) . The result is expressed as the inner product of certain symmetric functions. We are then led to a conjectural formula for the mixed Hodge polynomial, which is built out of (modified) Macdonald polynomials, their self-pairings, and self-pairings of wreath Macdonald polynomials.
In this article, we investigate the Gevrey regularity and the summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations with analytic coefficients at the origin of ℂ 2 . Given such a system, we first prove that the Gevrey regularity of its formal solutions follows a noteworthy dichotomy with respect to the s -Gevrey regularity of the inhomogeneity: for s ≥ s 0 with s 0 a nonnegative rational number entirely determined by a convenient Newton polygon attached to the given system, the solutions inherit the Gevrey regularity of the inhomogeneity; for s < s 0 , if any exists, the solutions keep the s 0 -Gevrey regularity defined by the structure of the initial system. Then, assuming s 0 > 0 and some convenient additional assumptions on the system, we give a necessary and sufficient condition for the 1 / s 0 -summability of the formal solutions in a given direction. Several examples illustrate these results.
Peral–Miyachi’s celebrated theorem states that the operator ( I - Δ ) - α 2 exp ( i - Δ ) is bounded on L p ( ℝ d ) if and only if α ≥ s p : = ( d - 1 ) 1 p - 1 2 . We extend this result to operators of the form ℒ = - ∑ j = 1 d a j + d ∂ j a j ∂ j , such that, for j = 1 , ⋯ , d , the functions a j and a j + d only depend on x j , are bounded above and below, but are merely Lipschitz continuous. This is below the C 1 , 1 regularity that is required in general situations. We construct spaces on which exp ( i ℒ ) is bounded by lifting L p functions to tent spaces, using wave packets adapted to the coefficients. The result then follows from Sobolev embedding properties of these spaces.
We study the sheaf of the locally square integrable holomorphic section of a vector bundle with semi-positive curved singular Hermitian metric. We confirm the coherence when its induced determinant metric has analytic singularities.
Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p -adic families of finite slope quaternionic automorphic forms over F . It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p -adic L -functions to triples of ordinary p -adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu–Zhang–Zhang to our families.
Let n ≥ 3 , and let Out ( W n ) be the outer automorphism group of a free Coxeter group W n of rank n . We study the growth of the dimension of the homology groups (with coefficients in any field 𝕂 ) along Farber sequences of finite-index subgroups of Out ( W n ) . We show that, in all degrees up to ⌊ n 2 ⌋ - 1 , these Betti numbers grow sublinearly in the index of the subgroup. When 𝕂 = ℚ , through Lück’s approximation theorem, this implies that all ℓ 2 -Betti numbers of Out ( W n ) vanish up to degree ⌊ n 2 ⌋ - 1 . In contrast, in top dimension equal to n - 2 , an argument of Gaboriau and Noûs implies that the ℓ 2 -Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Abért, Bergeron, Frączyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of W n has the homotopy type of a bouquet of spheres of dimension ⌊ n 2 ⌋ - 1 .
In this article, we provide a necessary and sufficient criterion for proper actions on ℍ n , n - 1 in terms of certain special Anosov representations in SO ( n , n ) . Moreover, we show that affine Anosov representations of any word hyperbolic group in SO ( n , n - 1 ) ⋉ ℝ 2 n - 1 are infinitesimal versions of such special Anosov representations. Finally, using the above two results we interpret Margulis spacetimes as infinitesimal versions of quotient manifolds of ℍ n , n - 1 . In the appendix, we give a description of the appropriate cross-ratios in our setting and their infinitesimal versions.
We survey different approaches to Suita’s conjecture and its various generalizations. We present a new and unified proof for generalized Suita conjectures with jets and weights, which is based on the concavity of certain minimal L 2 integrals and the necessary condition for linearity. Additionally, we provide some examples and counterexamples for the equalities in generalized Suita conjectures.
A meander is a pair consisting of a straight line in the plane and of a smooth closed curve transversally intersecting the line, considered up to an isotopy preserving the straight line. The number of meanders with 2 N intersections grows exponentially with N , but asymptotics still remains conjectural. A meander defines a pair of transversally intersecting simple closed curves on a 2 -sphere. In this paper we consider such pairs on a closed oriented surface of arbitrary genus. The number of these higher genus meanders still admits exponential upper and lower bounds as N grows. Fixing the number n of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus g meanders with n bigons and with at most 2 N intersections and show that it grows polynomially with N . We obtain a similar result in the case of oriented curves.
Let M be a hyperbolizable 3 -manifold with boundary, and let χ 0 ( M ) be a component of the PSL 2 ℂ -character variety of M that contains the convex co-compact characters. We show that the peripheral map i * : χ 0 ( M ) → χ ( ∂ M ) to the character variety of ∂ M is a birational isomorphism with its image, and in particular is generically a one-to-one map. This generalizes work of Dunfield (one cusped hyperbolic 3 -manifolds) and Klaff–Tillmann (finite volume hyperbolic 3 -manifolds). We use the Bonahon–Schläfli formula and volume rigidity of discrete co-compact representations.
The seminormalization of an algebraic variety is the biggest variety which is link to it with a birational, finite and bijective morphism. In this paper, we bring a new understanding to the seminormalization of complex algebraic varieties. We show that it can be obtained by replacing the structural sheaf of the variety by the sheaf of rational functions which extends continuously for the Euclidean topology. We further study this type of functions which can be seen as complex regulous functions, a class of functions recently introduced in real algebraic geometry, or as the algebraic counterpart of c-holomorphic functions.
We study the generalized classical Yang-Baxter equation (short: GCYBE) for central simple Lie algebras over fields of characteristic 0. Using a novel geometrization procedure, we assign to a solution of the GCYBE a cohomology free sheaf of Lie algebras on a projective curve. This assignment implies that all such solutions are algebraic in nature, i.e. they extend to rational functions on the product of two algebraic curves. Furthermore, the curves assigned to skew-symmetric solutions turn out to be either smooth, nodal, or cuspidal cubic plane curves. This results in a geometric trichotomy of skew-symmetric solutions of the GCYBE. Over the field of complex numbers, this geometric trichotomy implies the well-known Belavin-Drinfeld trichotomy, which states that skew-symmetric solutions of the GCYBE are either elliptic, trigonometric, or rational. As an interesting side result, we show that sheaves of Lie algebras with constant geometric fibers are locally free in the & eacute;tale topology. This is used to classify such sheaves on the complex plane with at most one puncture.
. Let C be a group acting on a CAT(0) space with contracting isometries. We study the random walk generated by an admissible measure on C. We prove that if the action is non-elementary and under optimal moment assumptions on the measure, the random walk satisfies a central limit theorem. The general our strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of the contact graph for the class of CAT(0) spaces. As a side result, we prove that the probability that the nth-step the random walk acts as a contracting isometry goes to 1 as n goes to infinity.
Let (K, v) be a valued field, the notions of augmented valuation, of limit augmented valuation and of admissible family of valuations enable to give a description of any valuation mu of K[x] extending v. In the case where the field K is algebraically closed, this description is particularly simple and we can reduce it to the notions of minimal pair and pseudo-convergent family. Let (K, v) be a henselian valued field and v the unique extension of v to the algebraic closure K(- )of K and let mu be a valuation of K[x] extending v, we study the extensions mu from mu to K[x] and we give a description of the valuations mu (-)(i )of K[x] which are the extensions of the valuations mu (-)(i) belonging to the admissible family associated with mu.
We classify which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that S(K) is bounded for each K. Moreover, if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).
For a symplectic vector space V, a projective subvariety Z ⊂ P V is a Legendrian variety if its affine cone Z⊂ V is Lagrangian. In addition to the classical examples of subadjoint varieties associated to simple Lie algebras, many examples of nonsingular Legendrian varieties have been discovered which have positive-dimensional automorphism groups. We give a characterization of subadjoint varieties among such Legendrian varieties in terms of the isotropy representation. Our proof uses some special features of the projective third fundamental forms of Legendrian varieties and their relation to the lines on the Legendrian varieties.
We give an affirmative answer to the Grunwald problem for new families of non-solvable finite groups G, away from the set of primes dividing |G|. Furthermore, we show that such G verify the condition (BM), that is, the Brauer-Manin obstruction to weak approximation is the only one for quotients of SL_n by G. These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups S_m, with m≠ 2,6, and alternating groups A_5. We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel's hypothesis.
Let G be a group and let 𝒢 be a free factor system of G, namely a free splitting of G as G=G_1*…*G_k*F_r. In this paper, we study the set of train track points for 𝒢-irreducible automorphisms ϕ with exponential growth (relatively to 𝒢). Such set is known to coincide with the minimally displaced set Min(ϕ) of ϕ. Our main result is that Min(ϕ) is co-compact, under the action of the cyclic subgroup generated by ϕ. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of Min(ϕ) is in uniform distance from Min(ϕ^-1). We also prove that the action of G on the product of the attracting and the repelling trees for ϕ, is discrete. Finally, we get some fine insight about the local topology of relative outer space. As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of Out(F_3) are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that Min(ϕ) is quasi-isometric to a line.
In this paper, we prove the Skoda-Zeriahi type integrability theorem with respect to some measure with L^1-density. In addition, we introduce the log-log threshold in order to detect singularities of Kähler potentials. We prove the positivity of the integrability threshold for such a measure and Kähler potentials with uniform log-log threshold. As an application, we prove the entropy compactness theorem for a family of potential functions of Poincaré type Kähler metrics with uniform log-log threshold. The Ohsawa-Takegoshi L^2-extension theorem and Skoda-Zeriahi's integrability theorem play a very important role in this paper.
