Abelian Image Research Articles

Abelian maps, brace blocks, and solutions to the Yang-Baxter equation

Let G be a finite nonabelian group. We show how an endomorphism of G with abelian image gives rise to a family of binary operations { ∘ n : n ∈ Z ≥ 0 } on G such that ( G , ∘ m , ∘ n ) is a skew left brace for all m , n ≥ 0 . A brace block gives rise to a number of non-degenerate set-theoretic solutions to the Yang-Baxter equation. We give examples showing that the number of solutions obtained can be arbitrarily large.

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

A Menagerie of SU(2)-Cyclic 3-Manifolds

Abstract We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image, respectively, among geometric 3-manifolds that are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds that do not admit degree-1 maps to any Seifert Fibered manifold other than $S^3$ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four $SU(2)$-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.

Low-dimensional representations of the three component loop braid group

Motivated by physical and topological applications, we study representations of the group LB3 of motions of 3 unlinked oriented circles in ℝ3. Our point of view is to regard the three strand braid group B3 as a subgroup of LB3 and study the problem of extending B3 representations. We introduce the notion of a standard extension and characterize B3 representations admitting such an extension. In particular we show, using a classification result of Tuba and Wenzl [Pacific J. Math. 197, 491–510 (2001)], that every irreducible B3 representation of dimension at most 5 has a (standard) extension. We show that this result is sharp by exhibiting an irreducible 6-dimensional B3 representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible 2-dimensional LB3 representations, (2) extensions of irreducible 3-dimensional B3 representations, and (3) irreducible LB3 representations whose restriction to B3 has abelian image.

Braid representations from unitary braided vector spaces

We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order. We verify this conjecture for the two infinite families of Gaussian and group-type braided vector spaces, as well as the generalization to quasi-braided vector spaces of group-type.

Erratum to: “Elementary embeddings in torsion-free hyperbolic groups”

The notations adopted are those of [Per11]. Proposition 5.11 of [Per11] states that if a torsion-free hyperbolic group A admits a cyclic JSJ-like decomposition Λ, and a non injective morphism f : A → A which restricts to conjugation on each non surface type vertex group, and sends surface type vertex groups to non abelian images, then there is a retraction r : A→ A′ which gives A a structure of hyperbolic floor over A′. Unfortunately, we realised that Proposition 5.11 fails to hold in a few exceptional low complexity cases. The natural modification to overcome this mistake is to proceed to a slight generalization of the notion of hyperbolic floors and hyperbolic towers, which we present in Section 1. As we will see in Section 2, however, this does not affect Theorem 1.2, the main result of the paper. Moreover, Theorem 1.2 is the only result which directly uses Proposition 5.11 in its proof. For a corrected version of the paper, see [Per09]. We sincerely apologize for any confusion caused by this mistake.

Inversion of a general hyperelliptic integral and particle motion in Hořava–Lifshitz black hole space-times

The description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of hyperelliptic integrals of all three kinds. The result of the inversion is defined locally, using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the θ-divisor. For a representation of the hyperelliptic functions the Klein–Weierstraß multi-variable σ-function is introduced. It is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and θ-constants. The cases of genus two, three, and four are considered in detail. The method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves. Applications are for instance solutions to the geodesic equations in the space-times of static, spherically symmetric Hořava–Lifshitz black holes.

Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity

The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space–times is reduced to the inversion of a holomorphic hyperelliptic integral. The result of the inversion is defined only locally, and is done using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the θ -divisor. For a representation of the hyperelliptic functions the Klein–Weierstraß multivariable sigma function is introduced. It is shown that all parameters needed for the calculations like period matrices and Abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and theta-constants. The cases of genus 2 and genus 3 are considered in detail. The method is exemplified by particle motion associated with a genus 3 hyperelliptic curve.

Bilinear recurrences and addition formulae for hyperelliptic sigma functions

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two. We show that the sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related to a Bäcklund transformation (BT) for an integrable Hamiltonian system, namely the discrete case (ii) Hénon-Heiles system.

$F$-split Galois representations are potentially abelian

In this note we relate the property of a semisimple -adic Galois representation being F-split to its having abelian image.

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D20 × Z2 (for example, if G ≃ D20 × K). (2) G has a normal 5-Sylow subgroup and an elementary abelian 2-Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5, or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2-Sylow subgroup S and 5-Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |ϕ(S)| = 8, and one of the following is true: (a) G = S × T and S is nonabelian; (b) G has a D10 image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in A Γ L(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations of the elementary abelian quotient of order 32 give a contradiction. In the third (due to an anonymous referee), the contradiction derives from a theorem of Lander together with Dillon's “dihedral trick.” Theorem B summarizes the open nonabelian cases based on this work. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 221–231, 2000

Ruling out (160, 54, 18) difference sets in some nonabelian groups

We prove the following theorems. Theorem A. Let G be a group of order 160 satisfying one of the following conditions. (1) G has an image isomorphic to D20 × Z2 (for example, if G ' D20 ×K). (2) G has a normal 5–Sylow subgroup and an elementary abelian 2–Sylow subgroup. (3) G has an abelian image of exponent 2, 4, 5 or 10 and order greater than 20. Then G cannot contain a (160, 54, 18) difference set. Theorem B. Suppose G is a nonabelian group with 2–Sylow subgroup S and 5–Sylow subgroup T and contains a (160, 54, 18) difference set. Then we have one of three possibilities. (1) T is normal, |φ(S)| = 8, and one of the following is true: (a) G = S ×T and S is nonabelian; (b) G has a D10 image; or (c) G has a Frobenius image of order 20. (2) G has a Frobenius image of order 80. (3) G is of index 6 in AΓL(1, 16). To prove the first case of Theorem A, we find the possible distribution of a putative difference set with the stipulated parameters among the cosets of a normal subgroup using irreducible representations of the quotient; we show that no such distribution is possible. The other two cases are due to others. In the second (due to Pott) irreducible representations ∗Support is gratefully acknowledged from the National Science Foundation Research Experiences for Undergraduates program, the Pew Charitable Trusts via the New England Consortium for Undergraduate Science Education, and the Howard Hughes Medical Foundation. Most of this work was done under Harriet Pollatsek’s supervision in the summer 1994 undergraduate mathematics research institute at Mount Holyoke College. The other authors are the (then) undergraduate researchers: Jason Alexander ’95, Lewis and Clark College, in the doctoral program in the philosophy and foundations of mathematics at UC Irvine; Rajalakshmi Balasubramanian ’96, Mount Holyoke College, in the doctoral program in statistics at Harvard University; Jeremy Martin ’96, Harvard University, in the doctoral program in mathematics at UC San Diego; Kimberley Monahan ’95, College of the Holy Cross, now teaching high school mathematics; Ashna Sen ’96, Mount Holyoke College, who completed a master’s degree in geophysics at Stanford University. 1 Ruling out (160,54,18) difference sets in some nonabelian groups (with Jason Alexander, Rajalakshmi Balasubramanian, Kimberly Monahan, Harriet Pollatsek, and Ashna Rubina Sen) J. Combin. Designs 8 (2000), no. 4, 221--231. Publisher's official version: http://dx.doi.org/10.1002/1520-6610(2000)8:4 3.0.CO;2-6, Open Access version: http://kuscholarworks.ku.edu/dspace/. [This document contains the author's accepted manuscript. For the publisher's version, see the link in the header of this document.]

The modular group algebra of certain finite groups with normal sylow p-subgroups

The modular group algebras we consider are of the form kP >a Q where P is a finite p-group, Q a finite p’ group, k a splitting field of characteristic p for Q, and the action of Q on P arises from a homomorphism a: Q + Aut P with abelian image A. (By the Schur-Zassenhaus theorem (cf. [3]), every finite group G with a normal Sylow p-subgroup P a G is a semidirect product P ~1 G/P; but of course the action need not be abelian.) Theorem 1.1 is a special case of Theorem 1.3, stated below, which gives sufficient conditions for an isomorphism of group algebras of the form kP >a Q. Theorem 1.3 in turn is an application of Theorem 1.2 which gives an explicit description of the blocks of such a group algebra. We note that we are dealing here with a class of blocks which do not in general have finite representation type. and indeed can have any p-group as defect group (the defect group is P). Theorem 1.2 can be regarded as lifting the Wedderburn theorem for kQ up to kP >a Q in accordance with the numerical data about blocks provided by Brauer’s theorem (cf. [ 1, p. 6341). It expresses kP ~1 Q as a direct sum of full

Growth sequences of finite groups

During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d(Gn) of the n-th direct power Gn of a non-trival finite group G tends to infinity with n. This has prompted me to ask the question: what are the ways in which the sequence {d(Gn)} can tend to infinity? Let us call this the growth sequence for G; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G'). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.