Cyclic Quotient Research Articles

Relationships between almost completely decomposable abelian groups and their multiplication groups

Let [Formula: see text] be the class of Abelian block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. In this paper, we study relationships between the above groups [Formula: see text] and their multiplication groups Mult [Formula: see text]. It is proved that groups from [Formula: see text] are determined by their multiplication groups. For a rigid group [Formula: see text], the isomorphism problem is solved: we describe multiplications from [Formula: see text] that define isomorphic rings on [Formula: see text]. We also describe Abelian groups that are realized as the multiplication group of some group in [Formula: see text]. We describe groups in [Formula: see text] that are isomorphic to their multiplication groups.

Full Exceptional Collections for Anticanonical Log del Pezzo Surfaces

Abstract Motivated by homological mirror symmetry, this paper constructs explicit full exceptional collections for the canonical stacks associated with the series of log del Pezzo surfaces constructed by Johnson and Kollár in [33]. These surfaces have cyclic quotient, non-Gorenstein, singularities. The construction involves both the $\operatorname {GL}(2,{\mathbb {C}})$ McKay correspondence, and the study of the minimal resolutions of the surfaces, which are birational to degree two del Pezzo surfaces. We show that a degree two del Pezzo surface arises in this way if and only if it admits a generalized Eckardt point, and in the course of the paper we classify the blow-ups of ${\mathbb {P}}^{2}$ giving rise to them. Our result on the adjoints of the functor of Ishii–Ueda [32] applies to any finite small subgroup of $\operatorname {GL}(2,{\mathbb {C}})$.

Experiments with Ceresa classes of cyclic Fermat quotients

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the L L -function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.

On the virtual and residual properties of a generalization of Bestvina-Brady groups

Previously one of us introduced a family of groups G^M_L(S), parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when G^M_L(S) is either residually finite or virtually torsion-free. In the case that M is a finite cover and S is periodic, there is an extension with kernel G_L^M(S) and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.

Cancellation for Surfaces Revisited

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × A n ≅ X ′ × A n X\times \mathbb {A}^n\cong X’\times \mathbb {A}^n for (affine) algebraic varieties X X and X ′ X’ implies that X ≅ X ′ X\cong X’ . In this paper we provide a criterion for cancellation by the affine line (that is, n = 1 n=1 ) in the case where X X is a normal affine surface admitting an A 1 \mathbb {A}^1 -fibration X → B X\to B with no multiple fiber over a smooth affine curve B B . For two such surfaces X → B X\to B and X ′ → B X’\to B we give a criterion as to when the cylinders X × A 1 X\times \mathbb {A}^1 and X ′ × A 1 X’\times \mathbb {A}^1 are isomorphic over B B . The latter criterion is expressed in terms of linear equivalence of certain divisors on the Danielewski-Fieseler quotient of X X over B B . It occurs that for a smooth A 1 \mathbb {A}^1 -fibered surface X → B X\to B the cancellation by the affine line holds if and only if X → B X\to B is a line bundle, and, for a normal such X X , if and only if X → B X\to B is a cyclic quotient of a line bundle (an orbifold line bundle). If X X does not admit any A 1 \mathbb {A}^1 -fibration over an affine base then the cancellation by the affine line is known to hold for X X by a result of Bandman and Makar-Limanov. If the cancellation does not hold then X X deforms in a non-isotrivial family of A 1 \mathbb {A}^1 -fibered surfaces X λ → B X_\lambda \to B with cylinders X λ × A 1 X_\lambda \times \mathbb {A}^1 isomorphic over B B . We construct such versal deformation families and their coarse moduli spaces provided B B does not admit nonconstant invertible functions. Each of these coarse moduli spaces has infinite number of irreducible components of growing dimensions; each component is an affine variety with quotient singularities. Finally, we analyze from our viewpoint the examples of non-cancellation constructed by Danielewski, tom Dieck, Wilkens, Masuda and Miyanishi, e.a.

Fujiki-Oka Resolution for Three-dimensional Cyclic Quotient Singularities via Binary Trees

For three dimensional cyclic quotient singularities of type 1r(1,a,r−a−1) (resp. type 1r(1,a,r−a)), the Fujiki-Oka resolution coincides with one of crepant resolutions (resp. an economic resolution). In this paper, we will characterize binary trees which gives the Fujiki-Oka resolution for the above two series of cyclic quotient singularities.

Degrees of faithful irreducible representations of metabelian groups

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

Stability for product groups and property (τ)

We study the notion of permutation stability (or P-stability) for countable groups. Our main result provides a wide class of non-amenable product groups which are not P-stable. This class includes the product group Σ×Λ, whenever Σ admits a non-abelian free quotient and Λ admits an infinite cyclic quotient. In particular, we obtain that the groups Fm×Zd and Fm×Fn are not P-stable, for any integers m,n≥2 and d≥1. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. The proof of our main result relies on a construction of asymptotic homomorphisms from Σ×Λ to finite symmetric groups starting from sequences of finite index subgroups in Σ and Λ with and without property (τ). Our method is sufficiently robust to show that the groups covered are not even flexibly P-stable, thus giving the first such non-amenable residually finite examples.

Direct Decomposition Theory of Torsion-Free Abelian Groups of Finite Rank: Graph Method

Graphical direct decomposition theory of block-rigid almost completely decomposable groups with cyclic regulator quotient is substantiated and effectively applied to construction of a group with predicted set of its direct decompositions up to near isomorphism.

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

FINITE GROUPS WITH A CYCLIC NORM QUOTIENT

The norm N(G) of a group G is the intersection of the nor- malizers of all the subgroups of G. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which as- serts that a finite group G is cyclic if and only if Aut(G)/P(G) is cyclic, where P(G) is the power automorphism group of G. The norm N(G) of a group G, first introduced by R. Baer (1) in 1934, is the intersection of the normalizers of all the subgroups of G. It is clear that N(G) is a characteristic subgroup of G and it contains the center Z(G). Also N(G) itself is a Dedekind group, and every element of N(G) induces a power automorphism on G. Many authors have investigated both N(G) and how N(G) influences the structure of G (see (2), (3), (4), (9), (10) and (11)). For instance, Schenkman proved that N(G) ≤ Z2(G) for any group G (see (9)), and Baer showed that a 2-group G must be a Dedekind group if the N(G) is nonabelian (see (2)). In (10) and (11), the authors have determined the structure of a finite group G satisfying |G : N(G)| = p or pq, where p,q are primes. In this paper, finite groups with a cyclic norm quotient are determined, see Section 2. Based on the above result, we will establish a characteristic of cyclic groups. Recall that a power automorphism of a group G is an automorphism that leaves every subgroup of G invariant. Under such an automorphism, every element of G is mapped to one of its powers. All power automorphisms of G constitute an abelian normal subgroup, denoted by P(G), of Aut(G). The structure of G and P(G) are strictly linked (for example, see (5), (12)), especially the quotient group Aut(G)/P(G) strongly influences the structure of G. In (11), groups satisfying |Aut(G)/P(G)| = 1,p or pq are completely clarified. In Section 3,

Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves

We obtain the trace map images of the values of certain harmonic volumes for some cyclic quotients of [Formula: see text]th Fermat curves. These provide the algorithm showing that the algebraic cycles, called [Formula: see text]th Ceresa cycles, are not algebraically equivalent to zero in their Jacobian varieties. We apply the method to curves for prime [Formula: see text] with [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text].

Torsion points on Jacobian varieties via Anderson’s $p$-adic soliton theory

Anderson introduced a $p$-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the $p$-adic loop group and the formal group.

Nilprogressions and groups with moderate growth

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (τ) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG.

Cyclic Regulator Quotient Groups in the Class of Almost Completely Decomposable Groups

Some results on almost completely decomposable groups with cyclic regulator quotient are discussed in the context of the general theory of almost completely decomposable Abelian groups.

Compact Kähler 3-manifolds without nontrivial subvarieties

We prove that any compact Kahler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classification of compact Kahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1, 1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider. In a similar vein, we show that a normal compact and Kahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no effective divisor. This is a crucial step towards completing the bimeromorphic classification of compact Kahler 3-folds.

Endomorphism Rings of Rigid Almost Completely Decomposable Abelian Groups

A necessary and sufficient condition is found for rigid almost completely decomposable Abelian groups with p-primary regulator quotients to have endomorphism rings isomorphic to those of the corresponding groups with cyclic regulator quotients.

Torsion Points on Hyperelliptic Jacobians via Anderson's $p$-adic Soliton Theory

We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation $y^2=x^{2g+1}+x$ with $g \geq 2$. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.

Symmetric Chain Decomposition for Cyclic Quotients of Boolean Algebras and Relation to Cyclic Crystals

The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.

INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.