Abelian Quotients Research Articles

Commensurators of thin normal subgroups and abelian quotients

Commensurators of thin normal subgroups and abelian quotients

The largest orbit sizes of linear group actions and abelian quotients

Let G be a finite group and let V be a finite and faithful G-module such that V is completely reducible, possibly of mixed characteristic, or V is a G-module over a field of characteristic p with Op(G)=1. Suppose that M is the largest orbit size in the action of G on V. It is known that the index of the commutator subgroup G′ in G is bounded by M, that is, |G:G′|≤M. In this paper, we classify all linear group actions in which |G:G′|=M. It turns out that our classification is a vast generalization of a classic 1967 result by D. Passman. We also include an application of the main results to Brauer's k(B)-problem.

Minimal generating sets in groups of $p$-automata

For an arbitrary odd prime $p$, we consider groups of all $p$-automata and all finite $p$-automata. We construct minimal generating sets in both the groups of all $p$-automata and its subgroup of finite $p$-automata. The key ingredient of the proof is the lifting technique, which allows the construction of a minimal generating set in a group provided a minimal generating set in its abelian quotient is given. To find the required quotient, the elements of the groups of $p$-automata and finite $p$-automata are presented in terms of tableaux introduced by L. Kaloujnine. Using this presentation, a natural homomorphism on the additive group of all infinite sequences over the field $\mathbb{Z}_p$ is defined and examined.

Abelian quotient of subgroups of solvable linear groups

In this paper, we study the relations of the sizes of abelian quotient of a subgroup of finite linear groups and the largest orbit size of the linear group actions. We study two conjectures raised by Keller and the author. We show that these two conjectures are not true in general, but also prove some special cases under the condition that the group is solvable.

Abelian quotients of the Y –filtration on the homology cylinders via the LMO functor

We construct a series of homomorphisms from the $Y$-filtration on the monoid of homology cylinders to torsion modules via the mod $\mathbb{Z}$ reduction of the LMO functor. The restriction of our homomorphism to the lower central series of the Torelli group does not factor through Morita's refinement of the Johnson homomorphism. We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements. We also determine the third graded quotient $Y_3\mathcal{IC}_{g,1}/Y_4$ of the $Y$-filtration.

Oriented pro-ell groups with the Bogomolov–Positselski property

For a prime number ell we say that an oriented pro-ell group (G,theta ) has the Bogomolov–Positselski property if the kernel of the canonical projection on its maximal theta -abelian quotient pi ^{mathrm {ab}}_{G,theta }:Grightarrow G(theta ) is a free pro-ell group contained in the Frattini subgroup of G. We show that oriented pro-ell groups of elementary type have the Bogomolov–Positselski property (cf. Theorem 1.2). This shows that Efrat’s Elementary Type Conjecture implies a positive answer to Positselski’s version of Bogomolov’s Conjecture on maximal pro-ell Galois groups of a field mathbb {K} in case that mathbb {K}^times /(mathbb {K}^times )^ell is finite. Secondly, it is shown that for an H^bullet -quadratic oriented pro-ell group (G,theta ) the Bogomolov–Positselski property can be expressed by the injectivity of the transgression map d_2^{2,1} in the Hochschild–Serre spectral sequence (cf. Theorem 1.4).

Torsion pairs and recollements of extriangulated categories

In this article, we prove that if is a recollement of extriangulated categories, then torsion pairs in and can induce torsion pairs in and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.

Abelian Quotients Arising from Extriangulated Categories via Morphism Categories

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let \((\mathcal {C},\mathbb {E},\mathfrak {s})\) be an extriangulated category with enough projectives \(\mathcal {P}\) and \({\mathscr{M}}\) be a full subcategory of \(\mathcal {C}\) containing \(\mathcal {P}\). We show that a certain quotient category of \(\mathfrak {s}\textup {-def}({\mathscr{M}})\), the category of \(\mathfrak {s}\)-deflations \(f {:} M_{1} {\rightarrow } M_{2}\) with \(M_{1},M_{2} {\in } {\mathscr{M}}\), is abelian. Our main theorem has two applications. If \({\mathscr{M}} {=} \mathcal {C}\), we obtain that a certain ideal quotient category \(\mathfrak {s}\textup {-tri}(\mathcal {C})/\mathcal {R}_{2}\) is equivalent to the category of finitely presented modules \(\textup {mod-}(\mathcal {C}/[\mathcal {P}])\), where \(\mathfrak {s}\)-tri\((\mathcal {C})\) is the category of all \(\mathfrak {s}\)-triangles. If \({\mathscr{M}}\) is a rigid subcategory, we show that \({\mathscr{M}}_{L}/[{\mathscr{M}}]\cong \textup {mod-}({\mathscr{M}}/[\mathcal {P}])\) and \({\mathscr{M}}_{L}/[{\Omega }{\mathscr{M}}]\cong (\textup {mod-}({\mathscr{M}}/[\mathcal {P}])^{\textup {op}})^{\textup {op}}\), where \({\mathscr{M}}_{L}\) (resp. \({\Omega }{\mathscr{M}}\)) is the full subcategory of \(\mathcal {C}\) of objects X admitting an \(\mathfrak {s}\)-triangle with \(M_{1}, M_{2}\in {\mathscr{M}}\) (resp. \(M\in {\mathscr{M}}\) and \(P\in \mathcal {P}\)). In particular, we have \(\mathcal {C}/[{\mathscr{M}}]\cong \textup {mod-}({\mathscr{M}}/[\mathcal {P}])\) and \(\mathcal {C}/[{\Omega }{\mathscr{M}}]\cong (\textup {mod-}({\mathscr{M}}/[\mathcal {P}])^{\textup {op}})^{\textup {op}}\) provided that \({\mathscr{M}}\) is a cluster-tilting subcategory.

A subexponential bound on the cardinality of abelian quotients in finite transitive groups

We show that, for every transitive group $G$ of degree $n\ge 2$, the largest abelian quotient of $G$ has cardinality at most $4^{n/\sqrt{\log_2 n}}$. This gives a positive answer to a 1989 outstanding question of L\'aszl\'o Kov\'acs and Cheryl Praeger.

Structure of normally and finitely non-co-Hopfian groups

A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G . We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: first, we show any proper self-embedding of G with finite-index characteristic image arises by pulling back an endomorphism of the abelianization; secondly, we prove special cases (for normal subgroups) of conjectures of Benjamini and Nekrashevych–Pete regarding the classification of scale-invariant groups.

Relative Growth in Hyperbolic Groups

In this note we obtain estimates on the relative growth of normal subgroups of non-elementary hyperbolic groups, particularly those with free abelian quotient. As a corollary, we deduce that the associated relative growth series fail to be rational.

Discriminantal bundles, arrangement groups, and subdirect products of free groups

We construct bundles $$E_k({{\mathscr {A}}},{{\mathscr {F}}}) \rightarrow M$$ over the complement M of a complex hyperplane arrangement $${{\mathscr {A}}}$$ , depending on an integer $$k \geqslant 1$$ and a set $${{\mathscr {F}}}=\{f_1, \ldots , f_\mu \}$$ of continuous functions $$f_i :M \rightarrow {{\mathbb {C}}}$$ whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence–Krammer–Bigelow representation of the pure braid group. We display such families $${{\mathscr {F}}}$$ for rank two arrangements, reflection arrangements of types $$A_\ell $$ , $$B_\ell $$ , $$D_\ell $$ , $$F_4$$ , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group $$\pi _1(M)$$ . We describe the kernel of the product $$\rho _{{\mathscr {X}}}:G \rightarrow \prod _{S \in {{\mathscr {X}}}} G_S$$ of homomorphisms of a finitely-generated group G onto quotient groups $$G_S$$ determined by a family $${{\mathscr {X}}}$$ of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections $$G \rightarrow G_S$$ split in a compatible way, we show the image of $$\rho _{{\mathscr {X}}}$$ is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of $$\pi _1(M)$$ , including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina–Brady groups, and a dichotomy for a decomposable arrangement $${{\mathscr {A}}}$$ : either $$\pi _1(M)$$ has a conjugation-free presentation or it is not residually nilpotent.

Abelian quotients of categories of short exact sequences

We mainly investigate abelian quotients of categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as module categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let (C,E) be an exact category. We denote by E(C) the category of bounded complexes whose objects are given by short exact sequences in E and by SE(C) the full subcategory formed by split short exact sequences. In general, E(C) is just an exact category, but the quotient E(C)/[SE(C)] turns out to be abelian. In particular, if (C,E) is Frobenius, we present three equivalent abelian quotients of E(C) and point out that the equivalences are actually given by left and right rotations. The abelian quotient E(C)/[SE(C)] admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results for triangulated versions.

Finite Groups With Few Relative Tensor or Exterior Degrees

A peculiar structure is present in a finite group G, when \(\mathcal {D}(G)=\{d(H,G) \ | \ H \ \text{ is } \text{ a } \text{ subgroup } \text{ of } \ G\}\) is small enough (here d(H, G) denotes the relative commutativity degree). Recent contributions show that G has elementary abelian quotients, when \(|\mathcal {D}(G)| \le 4\). We introduce a similar problem for the relative exterior degree \(d^\wedge (H,G)\) and for the relative tensor degree \(d^\otimes (H,G)\). Theorems of structure are shown when G has a small number of relative tensor (or exterior) degrees. Among other things, we give new estimations for the gap \(d^\wedge (H,G)-d^\otimes (H,G)\) and for the arithmetic average \((d^\wedge (H,G)+d^\otimes (H,G))/2\).

Abelian quotients of extriangulated categories

We prove that certain subquotient categories of extriangulated categories are abelian. As a particular case, if an extriangulated category $$\mathscr {C}$$ has a cluster-tilting subcategory $$\mathscr {X}$$ , then $$\mathscr {C}/\mathscr {X}$$ is abelian. This unifies a result by Koenig and Zhu (Math. Z. 258 (2008) 143–160) for triangulated categories and a result by Demonet and Liu (J. Pure Appl. Algebra 217(12) (2013) 2282–2297) for exact categories.

Abelian quotients and orbit sizes of linear groups

Let G be a finite group, and let V be a completely reducible faithful finite G-module (i.e., G ⩽ GL(V), where V is a finite vector space which is a direct sum of irreducible G-submodules). It has been known for a long time that if G is abelian, then G has a regular orbit on V. In this paper we show that G has an orbit of size at least |G/G′| on V. This generalizes earlier work of the authors, where the same bound was proved under the additional hypothesis that G is solvable. For completely reducible modules it also strengthens the 1989 result |G/G′| < |V| by Aschbacher and Guralnick.

Finite-dimensional representations constructed from random walks

Given a 1-cocycle b with coefficients in an orthogonal representation, we show that every finite dimensional summand of b is cohomologically trivial if and only if \| b(X_n) \|^2/n tends to a constant in probability, where X_n is the trajectory of the random walk (G,\mu) . As a corollary, we obtain sufficient conditions for G to satisfy Shalom's property H_{\mathrm{FD}} . Another application is a convergence to a constant in probability of \mu^{*n}(e) -\mu^{*n}(g) , n\gg m , normalized by its average with respect to \mu^{*m} , for any finitely generated infinite amenable group without infinite virtually abelian quotients. Finally, we show that the harmonic equivariant mapping of G to a Hilbert space obtained as an U -ultralimit of normalized \mu^{*n}- g \mu^{*n} can depend on the ultrafilter U for some groups.

Commutators in groups of piecewise projective homeomorphisms

In [7] Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and in [6] Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups H and G0. We show that the group H of piecewise projective homeomorphisms of R has the property that H″ is simple and that every proper quotient of H is metabelian. We establish simplicity of the commutator subgroup of the group G0, which admits a presentation with 3 generators and 9 relations. Further, we show that every proper quotient of G0 is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.

The mapping class group orbits in the framings of compact surfaces

AbstractWe compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.

Non-toric cones and Chern-Simons quivers

We obtain an integral formula for the volume of non-toric tri-Sasaki Einstein manifolds arising from nonabelian hyperkähler quotients. The derivation is based on equivariant localization and generalizes existing formulas for Abelian quotients, which lead to toric manifolds. The formula is particularly valuable in the context of AdS4 ×Y7 vacua of M-theory and their field theory duals. As an application, we consider 3d mathcal{N}=3 Chern-Simons theories with affine ADE quivers. While the  series corresponds to toric Y7, the widehat{D} and Ê series are non-toric. We compute the volumes of the corresponding seven-manifolds and compare to the prediction from supersymmetric localization in field theory, finding perfect agreement. This is the first test of an infinite number of non-toric AdS4/CFT3 dualities.