Finite Subgroup Research Articles

The automatic imprimitivity for Gq

Let Gq be the Drinfeld-Jimbo deformation of a connected simply-connected compact Lie group and T be its maximal torus. We show that the induction functors from the categories of T-equivariant correspondences to the categories of Gq-equivariant correspondences give equivalences of these categories under the existence of tracial states. As its application, we show some automatic imprimitivity results and classification results for extensions, subalgebras and finite index discrete quantum subgroups of Gqˆ.

Normal subalgebras of a polynomial ring

Let A be a finitely generated subalgebra of a polynomial ring k[x,y,z] over the complex field k. Assuming that A is normal, we clarify the structure of A under additional assumptions if dim⁡A≥2. If dim⁡A=2 and A is regular, then Spec A has an A1-fibration over P1 or A1 with restrictions on the number of multiple fibers (see Theorem 3). If dim⁡A=3, we assume that A is cofinite, i.e., k[x,y,z] is a finite A-module, and A contains a coordinate x. Then either A is a polynomial ring (see Theorem 6) or the invariant subring k[x,u,v]G with respect to a small finite subgroup G of GL(2,k) (see section 4). The results in section 4 are generalized in the next section.

On fillings of contact links of quotient singularities

AbstractWe study several aspects of fillings for links of general isolated quotient singularities using Floer theory, including co‐fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on the non‐existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

Finite normal subgroups of strongly verbally closed groups

Abstract In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.

On virtual indicability and property (T) for outer automorphism groups of RAAGs

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto \mathbb{Z} . We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.

Sharp bounds on the survival function of exchangeable min-stable multivariate exponential sequences

Abstract We derive lower and upper bounds for the survival function of an exchangeable sequence of random variables, for which the scaled minimum of each finite subgroup has a univariate exponential distribution. These bounds are sharp in the sense that both bounds themselves are attained by exchangeable sequences of the same kind, for which the (non-scaled) minimum of each subgroup has the same univariate exponential distribution as the original sequence. This result is equivalent to inequalities between infinite-dimensional stable tail dependence functions, which leads to inequalities between multivariate extreme-value copulas. In addition, it is explained how an infinite-dimensional symmetric stable tail dependence function can be obtained from its upper bound by censoring certain distributional information. This technique is applied to derive new parametric families.

COMPLETE EMBEDDINGS OF GROUPS

AbstractEvery countable group G can be embedded in a finitely generated group $G^*$ that is hopfian and complete, that is, $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of G. If G has a finite presentation (respectively, a finite classifying space), then so does $G^*$ . Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.

The Universe Inside Hall Algebras of Coherent Sheaves on Toric Resolutions

Abstract Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac–Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive nilpotent subalgebra. Given $\mathfrak {n}_{+}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_{3}(\mathbb {C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname {-}\mbox {Hilb}(\mathbb {C}^{3})$, and $\Psi : D^{b}_{G}(Coh(\mathbb {C}^{3}))\rightarrow D^{b}(Coh(Y))$ the derived equivalence of Bridgeland, King, and Reid. We present an (explicitly described) subset of objects in $Coh_{G}(\mathbb {C}^{3})$, s.t. the Hall algebra generated by their images under $\Psi $ is isomorphic to $U(\mathfrak {n}_{+})$. In case the field $\Bbbk $ (in place of $\mathbb {C}$) is finite and $\mbox {char}(\Bbbk )$ is coprime with the order of $G$, we establish isomorphisms of the corresponding “counting” Ringel–Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.

Kostant’s generating functions and Mckay–Slodowy correspondence

Let [Formula: see text] be a pair of finite subgroups of [Formula: see text] and [Formula: see text] a finite-dimensional fundamental [Formula: see text]-module. We study Kostant’s generating functions for the decomposition of the [Formula: see text]-module [Formula: see text] restricted to [Formula: see text] in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Abstract We consider the semi-direct products $G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$, and ${\mathbb{Z}}^{2}\rtimes \Gamma (2)$ (where $\Gamma (2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^{*}$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in $K_{0}$ provide explicit generators that are matched by the Baum–Connes assembly map.

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Слойная конечность в некоторых группах

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps

Let $X$ be a complex projective variety. Suppose that the group of birational automorphisms of $X$ contains finite subgroups isomorphic to $(\mathbb{Z}/N\mathbb{Z})^r$ for $r$ fixed and $N$ arbitrarily large. We show that $r$ does not exceed $2\dim(X)$. Moreover, the equality holds if and only if $X$ is birational to an abelian variety. We also show that an analogous result holds for groups of bimeromorphic automorphisms of compact Kähler spaces under some additional assumptions.

Steady Kähler–Ricci solitons on crepant resolutions of finite quotients of Cn$\mathbb {C}^n$

AbstractWe prove the existence of steady Kähler–Ricci solitons on equivariant crepant resolutions of , where is a finite subgroup of acting freely on .

Classification of irreducible admissible modules for 𝒮-local vertex operator algebra V #H

For an [Formula: see text]-local vertex operator algebra [Formula: see text], where [Formula: see text] is a simple vertex operator algebra, and [Formula: see text] is the group algebra of a finite subgroup of [Formula: see text], Frobenius reciprocity is investigated. We give an explicit construction and classification of admissible [Formula: see text]-modules in terms of admissible [Formula: see text]-modules. We also give a complete set of irreducible inequivalent admissible [Formula: see text]-modules.

On Locally Finite Subgroups in $ \operatorname{Lim}(N) $

On Locally Finite Subgroups in $ \operatorname{Lim}(N) $

An exposition of Jordan’s original proof of his theorem on finite subgroups of GLn(ℂ)

An exposition of Jordan’s original proof of his theorem on finite subgroups of GLn(ℂ)

Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups

Abstract By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.

Orthogonal irreducible representations of finite solvable groups in odd dimension

AbstractWe prove that if is a finite irreducible solvable subgroup of an orthogonal group with odd, then preserves an orthogonal decomposition of into 1‐spaces. In particular is monomial. This generalizes a theorem of Rod Gow.

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