Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from $G$ to $\Gamma$ as compositions of factorizations through $\Gamma$-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of $\Gamma$-limit groups as iterated generalized doubles over $\Gamma$.Comment: Corrected according to referee suggestions. Accepted to Groups, Complexity, Cryptolog

On factorizations of maps between curves

We examine the different ways of writing a cover of curves $\phi\colon C\to D$ over a field $K$ as a composition $\phi=\phi_n\circ\phi_{n-1}\circ\dots\circ\phi_1$, where each $\phi_i$ is a cover of curves over $K$ of degree at least $2$ which cannot be written as the composition of two lower-degree covers. We show that if the monodromy group $\textrm{Mon}(\phi)$ has a transitive abelian subgroup then the sequence $(\deg\phi_i)_{1\le i\le n}$ is uniquely determined up to permutation by $\phi$, so in particular the length $n$ is uniquely determined. We prove analogous conclusions for the sequences $(\textrm{Mon}(\phi_i))_{1\le i\le n}$ and $(\textrm{Aut}(\phi_i))_{1\le i\le n}$. Such a transitive abelian subgroup exists in particular when $\phi$ is tamely and totally ramified over some point in $D(\overline{K})$, and also when $\phi$ is a morphism of one-dimensional algebraic groups (or a coordinate projection of such a morphism). Thus, for example, our results apply to decompositions of polynomials of degree not divisible by $\textrm{char}(K)$, additive polynomials, elliptic curve isogenies, and Latt\`es maps.Comment: 23 page

Nonuniqueness of semidirect decompositions for semidirect products with directly decomposable factors and applications for dihedral groups

Nonuniqueness of semidirect decompositions of groups is an insufficiently studied question in contrast to direct decompositions. We obtain some results about semidirect decompositions for semidirect products with factors which are nontrivial direct products. We deal with a special case of semidirect product when the twisting homomorphism acts diagonally on a direct product, as well as for the case when the extending group is a direct product. We give applications of these results in the case of generalized dihedral groups and classic dihedral groups $D_{2n}$. For $D_{2n}$ we give a complete description of semidirect decompositions and values of minimal permutation degrees

Abelian Splittings and JSJ-Decompositions of Bestvina--Brady Groups

We give a characterization of Bestvina--Brady groups split over abelian subgroups and describe a JSJ-decomposition of Bestvina--Brady groups.Comment: Fix the proof of Theorem 3.

Quasiprimitive groups and blow-up decompositions

The blow-up construction by L. G. Kov\'acs has been a very useful tool to study embeddings of finite primitive permutation groups into wreath products in product action. In the present paper we extend the concept of a blow-up to finite quasiprimitive permutation groups, and use it to study embeddings of finite quasiprimitive groups into wreath products

Decompositions of singular abelian surfaces

Given an abelian surface, the number of its distinct decompositions into a product of elliptic curves has been described by Ma. Moreover, Ma himself classified the possible decompositions for abelian surfaces of Picard number $1 \leq \rho \leq 3$. We explicitly find all such decompositions in the case of abelian surfaces of Picard number $\rho= 4$. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decomposition. We also provide an alternative and simpler proof of Ma's formula, and an application to singular K3 surfaces.Comment: 30 pages. Final version. Comments are still very welcome

Three types of inclusions of innately transitive permutation groups into wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also show how to construct examples of inclusions for each type.Comment: v1 is replaced after minor alterations not concerning maths conten

Hamilton decompositions of one-ended Cayley graphs

We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a decomposition into $n$ edge-disjoint Hamiltonian double-rays for all $n \in \mathbb{N}$.Comment: 17 pages, 4 figure

Computing algorithm for reduction type of CM abelian varieties

Let $\mathcal{A}$ be an abelian variety over a number field, with a good reduction at a prime ideal containing a prime number $p$. Denote by ${\rm A}$ an abelian variety over a finite field of characteristic $p$, obtained by the reduction of $\mathcal{A}$ at the prime ideal. In this paper we derive an algorithm which allows to decompose the group scheme ${\rm A}[p]$ into indecomposable quasi-polarized ${\rm BT}_1$-group schemes. This can be done for the unramified $p$ on the basis of its decomposition into prime ideals in the endomorphism algebra of ${\rm A}$. We also compute all types of such correspondence for abelian varieties of dimension up to $5$. As a consequence we establish the relation between the decompositions of prime $p$ and the corresponding pairs of $p$-rank and $a$-number of an abelian variety ${\rm A}$.Comment: arXiv admin note: text overlap with arXiv:1209.520

Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.Comment: 13 page