Finite Subgroup Research Articles

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 $\operatorname {GL}(2,{\mathbb {C}})$.

Algebraic and context-free subsets of subgroups

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.

Invariant smooth quartic surfaces by all finite primitive subgroups of [formula omitted

For each finite primitive subgroup G of PGL4(C), we find all the smooth G-invariant quartic surfaces. We also find all the faithful representations in PGL4(C) of the smooth quartic G-invariant surfaces by the groups: A5,S5,PSL2(F7), A6, Z24⋊Z5 and Z24⋊D10. The primitive representation of these groups is precisely the subgroups of PGL4(C) for which P3 is not G-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by {x04+x14+x24+x34+12x0x1x2x3=0} (unique up to projective equivalence).

Sumsets in dicyclic groups Q 4 n and Um,n

Let G be an arbitrary group. For any subsets A and B of G, let A * B = { a * b ; a ∈ A , b ∈ B } where ` * ’ is the binary operation on G. By μ G ( r , s ) , we denote the minimum cardinality of the set A * B , where | A | = r and | B | = s . In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers r , s with 1 ≤ r , s ≤ g , k G ( r , s ) ≤ μ G ( r , s ) , where k G ( r , s ) = min d ∈ H ( G ) { ( ⌈ r d ⌉ + ⌈ s d ⌉ − 1 ) d } and H ( G ) is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group D q of index q = p v . In this paper, we determine the upper bound for the size of μ Q 4 n ( r , s ) and μ U m , n ( r , s ) , where Q 4 n = 〈 a , b : a 2 n = e , b 2 = a n , a b = b a − 1 〉 is a dicyclic group of order 4n and U m , n = 〈 a , b : a 2 n = e , b m = e , a b a − 1 = b − 1 〉 and verify the conjecture proposed by Eliahou et al. [7] for Q 4 n for n to be a prime power.

Orbit Chern classes

Let G ⊆ GL ( n , F ) be a finite subgroup G of the general linear group over a field F . Assume that F is algebraically closed. We show that the associated ring of polynomial invariants is characterized by its orbit Chern classes and vice versa. Moreover, the associated ring of polynomial invariants is characterized by its orbits on the dual space V * = ( F n ) * and vice versa.

Equivariant pliability of the projective space

We classify finite subgroups Gsubset {textrm{PGL}}_4({mathbb {C}}) such that {mathbb {P}}^3 is not G-birational to conic bundles and del Pezzo fibrations, and explicitly describe all G-Mori fibre spaces that are G-birational to {mathbb {P}}^3 for these subgroups.

Correction to: On the finite index subgroups of Houghton’s groups

This note resolves an issue raised by Prof. Derek Holt for the paper “On the finite index subgroups of Houghton’s groups”.

Finite index subgroups in Chevalley groups are bounded: An addendum to “On Bi-Invariant Word Metrics”

We prove that finite index subgroups in [Formula: see text]-arithmetic Chevalley groups are bounded.

A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Let G G be a finite subgroup of S U ( 4 ) \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K K -theoretic stable pair invariants on a crepant resolution of the affine quotient C 4 / G {\mathbb {C}}^4/G , and conjecture a closed formula for their generating series in terms of the root system of G G . In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [ C 4 / Z r ] [{\mathbb {C}}^4/{\mathbb {Z}}_r] , [ C 4 / Z 2 × Z 2 ] [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2] . Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

Finite Groups Acting on Compact Complex Parallelizable Manifolds

Abstract We prove that the automorphism group of a compact complex parallelizable manifold is Jordan. In the course of the proof, we show that outer automorphism groups of cocompact lattices in complex Lie groups have bounded finite subgroups.

On the geometry of a Picard modular group

We study geometric properties of the action on the complex hyperbolic plane H^2_\mathbb{C} of the Picard modular group \Gamma=\mathrm{PU}(2,1,\mathcal{O}_7) , where \mathcal{O}_7 denotes the ring of algebraic integers in \mathbb{Q}(i\sqrt{7}) . We list conjugacy classes of maximal finite subgroups in \Gamma and give an explicit description of the Fuchsian subgroups that occur as stabilizers of mirrors of complex reflections in \Gamma . As an application, we describe an explicit torsion-free subgroup of index 336 in \Gamma .

Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the Johnson kernel

Let \mathrm{ IA}_n denote the group of \mathrm{ IA} -automorphisms of a free group of rank n , and let \mathcal I_n^b denote the Torelli subgroup of the mapping class group of an orientable surface of genus n with b boundary components, b=0,1 . In 1935, Magnus proved that \mathrm{ IA}_n is finitely generated for all n , and in 1983, Johnson proved that \mathcal I_n^b is finitely generated for n\geq 3 . It was recently shown that for each k\in{\mathbb N} , the k -th terms of the lower central series \gamma_k \mathrm{ IA}_n and \gamma_k\mathcal I_n^b are finitely generated when n\gg k ; however, no information about finite generating sets was known for k>1 . The main goal of this paper is to construct an explicit finite generating set for \gamma_2 \mathrm{ IA}_n = [\mathrm{ IA}_n,\mathrm{ IA}_n] and almost explicit finite generating sets for \gamma_2\mathcal I_n^b and the Johnson kernel, which contains \gamma_2\mathcal I_n^b as a finite index subgroup.

Families of ϕ‐congruence subgroups of the modular group

AbstractWe introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ‐congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi‐unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi‐unipotent case, we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve defined by the commutator subgroup of the modular group.

Commensurations of the outer automorphism group of a universal Coxeter group

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank n , which is the free product W_n of n copies of \mathbb{Z}/2\mathbb{Z} . We prove that for n\geq 5 the natural map \mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n)) is an isomorphism and that every isomorphism between finite index subgroups of \mathrm{Out}(W_n) is given by a conjugation by an element of \mathrm{Out}(W_n) .

Hamiltonians and gauge-invariant Hilbert space for lattice Yang-Mills-like theories with finite gauge group

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite gauge group, for example a finite subgroup of a compact Lie group. We show that the electric Hamiltonian admits an interpretation as a certain natural, nonunique Laplacian operator on the finite Abelian or non-Abelian group and derive some consequences from this fact. Independent of the chosen Hamiltonian, we provide a full explicit description of the physical, gauge-invariant Hilbert space for pure gauge theories and derive a simple formula to compute its dimension. We illustrate the use of the gauge-invariant basis to diagonalize a dihedral gauge theory on a small periodic lattice.

ADHM in 8d, coloured solid partitions and Donaldson-Thomas invariants on orbifolds

We study the moduli space of SU(4) invariant BPS conditions in supersymmetric gauge theory on non-commutative C4 by means of an ADHM-like quiver construction and we classify the invariant solutions under the natural toric action in terms of solid partitions. In the orbifold case C4/G, G being a finite subgroup of SU(4), the classification is given in terms of coloured solid partitions. The statistical weight for their counting is defined through the associated equivariant cohomological gauge theory. We explicitly compute several terms of the partition function expansion in the instanton counting parameter on C4 and C2×(C2/Z2), which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.

Finite p-Irregular Subgroups of $${\text {PGL}}_2(k)$$

In the late 19th century, Klein inaugurated a program for describing the finite subgroups of $${\text {PGL}}_2(k)$$ by treating the case in which the field k is the complex numbers. Gierster and Moore extended Klein’s arguments to deal with finite fields. In the past century, additional contributions to this problem were made by Serre, Suzuki, and Beauville, among others. We complete this program by giving a classification of the finite subgroups of $${\text {PGL}}_2(k)$$ with order divisible by p, up to conjugation, for an arbitrary field k of positive characteristic p.

On the Structure of the Mislin Genus of a Pullback

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set G(Q) if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations Qp and Rp are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback Ht from the l-equivalences Hi→H and Hj→H, t≡(i+j)mods, where s=|G(H)|, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product G×K of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.

ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

AbstractThe aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$ , with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case.

An inequality for non-microstates free entropy dimension for crossed products by finite abelian groups

For certain generating sets of the subfactor pair M\subset M\rtimes G where G is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Schreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty} for a large class of algebras M .