Hurwitz Equivalence in Tuples of Dihedral Groups, Dicyclic Groups, and Semidihedral Groups

Let D[subscript 2N] be the dihedral group of order 2N, Dic[subscript 4M] the dicyclic group of order 4M, SD[subscript 2m] the semidihedral group of order 2[superscript m], and M[subscript 2m] the group of order 2[superscript m] with presentation M[subscript 2m] = ⟨α,β∣α[superscript 2m−1] = β[superscript 2] = 1, βαβ[superscript −1] = α[superscript 2m−2+1]⟩. We classify the orbits in D[n over 2N], Dic[n over 4M], SD[n over 2m], and M[n over 2m] under the Hurwitz action.National Science Foundation (U.S.) (Grant DMS 0754106)United States. National Security Agency (Grant H98230-06-1-001)Massachusetts Institute of Technology. Department of Mathematic

Hurwitz Equivalence in Dihedral Groups

In this paper we determine the orbits of the braid group B[subscript n] action on G[superscript n] when G is a dihedral group and for any T ∈ G[superscript n]. We prove that the following invariants serve as necessary and sufficient conditions for Hurwitz equivalence. They are: the product of its entries, the subgroup generated by its entries, and the number of times each conjugacy class (in the subgroup generated by its entries) is represented in T

Connectivity Properties of Factorization Posets in Generated Groups

We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.Comment: 35 pages, 17 figures. Comments are very welcome. Final versio

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

We prove that two reflection factorizations of a parabolic quasi-Coxeter element in a finite Coxeter group belong to the same Hurwitz orbit if and only if they generate the same subgroup and have the same multiset of conjugacy classes. As a lemma, we classify the finite Coxeter groups for which every reflection generating set that is minimal under inclusion is also of minimum size.Comment: 10 pages, 1 figure, comments very much welcome

Irreducibility of the space of dihedral covers of the projective line of a given numerical type

We show in this paper that the set of irreducible components of the family of Galois coverings of P^1_C with Galois group isomorphic to D_n is in bijection with the set of possible numerical types. In this special case the numerical type is the equivalence class (for automorphisms of D_n) of the function which to each conjugacy class \mathcal{C} in D_n associates the number of branch points whose local monodromy lies in the class \mathcal{C}.Comment: 18 pages, to appear in Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., volume in memory of Giovanni Prod

Genus stabilization for the components of moduli spaces of curves with symmetries

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant \e for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant \e, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H_2(G, \ZZ). We achieve our result showing that the stable classes are in bijection with the set of admissible classes \e

The irreducible components of the moduli space of dihedral covers of algebraic curves

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an effective action by the dihedral group $D_n$. This invariant has been used in the meanwhile by the authors in order to extend the genus stabilization result of Livingston and Dunfield and Thurston to the ramified case. This new version contains an appendix clarifying the correspondence between the above components and the image loci in the moduli space M_g (classifying when two such components have the same image).Comment: 37 pages, final version appearing in 'Groups, Geometry and Dynamics