We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the points of the fiber. This turns out to be the same problem as the action of pi_1(P_{2g+6}) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P^1 with deck group (Z/N)^2:S_3.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
This article is a continuation of the article with the same title (see arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and having fixed monodromy type $t$ consisting of a collection of local monodromy types (that is, a collection of conjugacy classes of permutations $σ$ of the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,...,d\}$). We prove that if the type $t$ contains big enough number of local monodromies belonging to the conjugacy class $C$ of an…
Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth, projective curve Y of positive genus, simply branched in n points, with full monodromy group S_d (math.AG/0205056). We sharpen this result and prove that H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y if n >= max{2,2d-6}. We extend the result to coverings simply branched in all but one point of the discriminant. Fixing the ramification multiplicities over the special point we prove that the corresponding Hurwitz sp…
Motivated by the problem of Hurwitz equivalence of $Δ^2$ factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, obtained by projecting the $Δ^2$ factorizations into $S_n$. We get $1_{S_n}$ factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very …
We describe an algorithm that, given a k-tuple of permutations representing the monodromy of a rational map, constructs an arbitrarily precise floating-point complex approximation of that map. We then explain how it has been used to study a problem in dynamical systems raised by Cui.
We prove the irreducibility of the Hurwitz spaces which parametrize Galois coverings of P^1 whose Galois group is an arbitrary Weyl group and the local monodromies are reflections. This generalizes a classical theorem due to Clebsch and Hurwitz.
We investigate the Hurwitz action of the braid group Br_n on the n-fold Cartesian product of Br_3 and determine some stabilisers of its Artin systems. Our algebraic result is complemented by a geometric study of families of plane polynomial coverings of degree 3. Together they lead to characterisations of the set of 'paths realised by degenerations' of the polynomials as defined by Donaldson in 'Polynomials, vanishing cycles and Floer homology'.
For a positively presented group G, we provide a criterion for two tuples of positive group generators of G to be Hurwitz equivalent or Hurwitz-conjugation equivalent. We also present an algorithmic approach to solve the Hurwitz equivalence and the Hurwitz search problems by using the word reversing method.
In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $Δ_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces. The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid gr…
This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in $PGL_n(F)$ for $n\leq 7$ which are Hurwitz, where $F$ is an algebraically closed field. For the groups $G_2(q)$, $q\geq 5$, and the Janko groups $J_1$ and $J_2$ it provides explicit $(2,3,7)$-generators.
If $\mathcal S$ is a Beauville surface $({\mathcal C}_1\times{\mathcal C}_2)/G$, then the Hurwitz bound implies that $|G|\le 1764\,χ({\mathcal S})$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves ${\mathcal C}_i$. Equivalently, $G$ has two generating triples of type $(2,3,7)$, such that no generator in one triple is conjugate to a power of a generator in the other. We show that this property is satisfied by alternating groups $A_n$, their double covers $2.A_n$, and special linear groups $SL_n(q)$ if $n$ is sufficiently large, but by no sporadic simp…
In 1963, Greenberg proved that every finite group appears as the monodromy group of some morphism of Riemann surfaces. In this paper, we give two constructive proofs of Greenberg's result. First, we utilize free groups, which given with the universal property and their construction as discrete subgroups of $\operatorname{PSL}_2(\mathbb{R})$, yield a very natural realization of finite groups as monodromy groups. We also give a proof of Greenberg's result based on triangle groups $Δ(m, n, k)$. Given any finite group $G$, we make use of subgroups of $Δ(m, n, k)$ in order to explicitly find a morp…
We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d, 1, n)$ (the group of $d$-colored permutations) and $G(d, d, n)$.
We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.
There are natural actions of the braid groups on the products of the braid groups, called the Hurwitz action. We first study the roots of centralizers in the braid groups. By using the structure of the roots, we provide a criterion for the Hurwitz orbit become finite and give an upper bound of the size of a finite orbit in length 2 or degree 3 case.
In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. In the fourth and fifth sections, we specialize to quite special classses of hypergeomtric sheaves. We give relatively "simple" formulas for their trace functions, and a criterion for them to have finite monodromy. In the next section, we prove that three of them have finite monodromy groups.We then give some results on finite complex linear groups. We next use these group theoretic results to show that one of our local systems, of rank $24…
We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.
It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (respectively, any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that there exists a pseudo-holomorphic curve (a Hurwitz curve) in $\mathbb C\mathbb P^2$ whose fundamental group of the complement is not Hopfian and, respectively, this group is not residually finite. In addition, it is proved that there exist an irreducible nonsingular algebraic curve $C\subset \mathbb C^2$ and a bi-disk $D\subset \mathbb C^2$ such that the fun…
We prove that the exceptional group $E_6(2)$ is not a Hurwitz group. In the course of proving this, we complete the classification up to conjugacy of all Hurwitz subgroups of $E_6(2)$, in particular, those isomorphic to $L_2(8)$ and $L_3(2)$.
For a given group $G$ the orientably regular maps with orientation-preserving automorphism group $G$ are used as the vertices of a graph $Ø(G)$, with undirected and directed edges showing the effect of duality and hole operations on these maps. Some examples of these graphs are given, including several for small Hurwitz groups. For some $G$, such as the affine groups ${\rm AGL}_1(2^e)$, the graph $Ø(G)$ is connected, whereas for some other infinite families, such as the alternating and symmetric groups, the number of connected components is unbounded.
We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.
A Hurwitz generating triple for a group $G$ is an ordered triple of elements $(x,y,z) \in G^3$ where $x^2=y^3=z^7=xyz=1$ and $\langle x,y,z \rangle = G$. For the finite quasisimple exceptional groups of types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$, we provide restrictions on which conjugacy classes $x$, $y$ and $z$ can belong to if $(x,y,z)$ is a Hurwitz generating triple. We prove that there exist Hurwitz generating triples for $F_4(3)$, $F_4(5)$, $F_4(7)$, $F_4(8)$, $E_6(3)$ and $E_7(2)$, and that there are no such triples for $F_4(2^{3n-2})$, $F_4(2^{3n-1})$, $E_6(7^{3n-2})$, $E_6(7^{3n-1})…
In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $Δ_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are Hurwitz equivalent. This theorem is a generalization of the theorem we have proved in \cite{Tz}, using an algebraic approach unlike the proof in \cite{Tz} which is geometric. The application of the result is applied to the classification of algebraic surfaces up to deformation. …
We first develop some basic facts about certain sorts of rigid local systems on the affine line in characteristic $p>0$. We then apply them to exhibit a number of rigid local systems of rank $23$ on the affine line in characteristic $p=3$ whose arithmetic and geometric monodromy groups are the Conway group $\mathrm{Co}_3$ in its orthogonal irreducible representation of degree $23$.
We prove that the universal covering of an alternating group Alt(n) which is Hurwitz is still Hurwitz, with 31 exceptions, 30 of which are detectable by the genus formula.
We first develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic $p>0$. One of these, of rank $23$ in characteristic $p=3$, turns out to have the Conway group $\mathrm{Co}_2$, in its irreducible orthogonal representation of degree $23$, as its arithmetic and geometric monodromy groups.
Hurwitz orbits are the orbits of the braid group action on the powers of a rack. Hurwitz orbits for the action of the braid group on three strands are used in \cite{21} and \cite{22} for the classification of Nichols algebras. This classification is based on a combinatorial invariant called plague on the Hurwitz orbits. The method to calculate plagues on the Hurwitz orbits is formulated in \cite{22} by using a cellular automaton on the Hurwitz orbits and their quotients. By using this cellular automaton-based method we estimate the minimal plagues on the Hurwitz orbits.
We provide a variety of cases in which two factorizations have Hurwitz orbits of the same size. We begin with prototypical results about factorizations of length two, and show that cycling elements or flipping and inverting elements in any factorization preserves Hurwitz orbit size. We prove that "double reverse" factorizations in groups with special presentations have Hurwitz orbits of equal size, and end with applications to complex reflection groups.
The article [14] gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomials, and for which, the absolute value of the leading coefficient of the difference polynomial is greater than 2. It follows from [12] and [14] that 12 of the 51 monodromy groups are arithmetic (cf. Table 1); and the thinness of 13 of the remaining 39 monodromy groups follows from [3] (cf. Table 2). In this article, we show that 15 of the remaining 26 monodromy groups are arithmetic (cf. Table 3).
This paper discusses iterated monodromy groups for transcendental functions. We show that for every post-singularly finite entire transcendental function, the iterated monodromy action can be described by bounded activity automata of a special form, called "dendroid automata". In particular, we conclude that the iterated monodromy group of a post-singularly finite entire function is amenable if and only if the monodromy group is.
We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an invariant known as a "n-spin structure" constrains the image in ways not predicted by previous work of Beauville. Second, we show that for degree d=5, our invariant is the only obstruction for a mapping class to be contained in the image. This requires combining the algebro-geome…
We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.
In [12], we show that 3 of the 14 hypergeometric monodromy groups associated to Calabi-Yau threefolds, are arithmetic. Brav-Thomas (in [3]) show that 7 of the remaining 11, are thin. In this article, we settle the arithmeticity problem for the 14 monodromy groups, by showing that, the remaining 4 are arithmetic.
We classify the finite primitive permutation groups which have a cyclic subgroup with two orbits. This extends classical topics in permutation group theory, and has arithmetic consequences. By a theorem of C. L. Siegel, affine algebraic curves with infinitely many integral points are parametrized by rational functions whose monodromy groups have this property. We classify the possibilities of these monodromy groups, and give an application to Hilbert's irreducibility theorem.
The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.
The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m, p, n)$ of complex ref…
In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants
Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of $z^2+i$. This is the first non-trivial example supporting the conjecture.
We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters $\left(\tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right)$ at infinity and maximal unipotent monodromy at zero, for any integer $N\geq 4$. Additionally, we relate the cones used for ping-pong in $\mathbb R^4$ with crooked surfaces, which we then use to exhibit do…
We study the action of the group of polynomial automorphisms of C^n (n>2) which preserve the Markoff-Hurwitz polynomial H(x):= x_1^2 + x_2^2 + ... + x_n^2 - x_1 x_2 ... x_n. Our main results include the determination of the group, the description of a non-empty open subset of C^n on which the group acts properly discontinuously (domain of discontinuity), and identities for the orbit of points in the domain of discontinuity.
This paper introduces iterated monodromy groups for transcendental functions and discusses them in the simplest setting, for post-singularly finite exponential functions. These groups are self-similar groups in a natural way, based on an explicit construction in terms of kneading sequences. We investigate the group theoretic properties of these groups, and show in particular that they are amenable, but they are not elementary subexponentially amenable.
We present all Belyi maps P^1(C) -> P^1(C) having almost simple primitive monodromy groups (not isomorphic to A_n or S_n) containing rigid and rational generating triples of degree between 50 and 250. This also leads to new polynomials having almost simple Galois groups over Q(t).
We describe the iterated monodromy groups associated with post-critically finite quadratic polynomials, and explicit their connection to the `kneading sequence' of the polynomial. We then give recursive presentations by generators and relations for these groups, and study some of their properties, like torsion and `branchness'.
We associate a group $IMG(f)$ to every covering $f$ of a topological space $M$ by its open subset. It is the quotient of the fundamental group $π_1(M)$ by the intersection of the kernels of its monodromy action for the iterates $f^n$. Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of $f$ is related to the group. In particular, the Julia set of $f$ can be reconstructed from $\img(f)$ (from its action on the tree), if $f$ is expanding.
We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.
We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis and the question whether there is an anlogue of the well known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis and Reiner as well a…
This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations.
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
This is an overview of recent results on the use of 2D Toda $τ$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $τ$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\…
