Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

Panov, Taras; Veryovkin, Yakov

doi:10.1070/sm8701

Cited by 32 publications

(46 citation statements)

References 22 publications

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“…Related results, [51,114], identify π 1 Z(K; (RP ∞ , * )) as a right-angled Coxeter groups RC K . From this follows the fact that π 1 Z(K; (D 1 , S 0 )) = [RC K , RC K ], the commutator subgroup.…”

Section: −→←−mentioning

confidence: 90%

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

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A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.The paper is organized as follows. A brief technical discussion of the origin of polyhedral products as moment-angle complexes in Sections 2 and 3, includes the original topological construction of toric manifolds by M. Davis and T. Januszkiewicz [52] and the reformulation of one of their main spaces as a moment-angle complex by V. Buchstaber and T. Panov, [34,35].A discussion follows in Section 4 about the independent discovery of moment-angle complexes as intersections of certain quadrics. We describe briefly the work of S. López de Medrano [100], S. López de Medrano and A. Verjovsky [101], F. Bosio and L. Meersseman [30], followed by that of S. López de Medrano and S. Gitler [65]. This approach has proved effective in identifying classes of moment-angle manifolds that are connected sums of products of spheres. This is followed in Section 5 by a sketch of the computation of the cohomology ring of moment-angle complexes.We begin our focus on more general polyhedral products in Section 6, by describing their behaviour with respect to the exponentiation of CW-pairs. This has led to an application to the study of toric manifolds [58,128,129,124,70,72,45], and orbifolds [18]. An application to topological joins is included as an example of the utility of this property with respect to exponentiation.The behaviour of polyhedral products with respect to fibrations, due to G. Denham and A. Suciu [55], is surveyed briefly in Section 7. We take advantage of the formalism to describe briefly the early work in this area by G. Porter [117,118], T. Ganea [64] and A. Kurosh [92]. and G. W. Whitehead. Instances of polyhedral products appear also in the work of D. Anick [5].In Section 8, we review the various fundamental unstable and stable splitting theorems for the polyhedral product. These theorems give access to the homotopy type of the polyhedral product in many of the most important cases and drive a number of applications. The identification of the various wedge summands which appear in the stable decompositions, occupies the second part of this section.A fine theorem of A. Al-Raisi [2] shows that, in certain cases, the stable splitting of the polyhedral product can be chosen to be equivariant with respect to the action of Aut(K). This and other related results, described in Sec...

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Section: −→←−mentioning

confidence: 90%

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

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“…Proof. As proved in [PV16], for a right-angled Coxeter group G, the commutator subgroup G is free group if and only if the Coxeter graph of G, Γ G is chordal.…”

Section: Further Eliminationmentioning

confidence: 91%

“…Even if a group is finitely generated, it is a non-trivial problem to compute its rank, that is the smallest cardinality of a generating set for the group. In [PV16], Panov and Verëvkin constructed classifying spaces for the commutator subgroups of right-angled Coxeter groups and have given a general formula for the rank of such groups, see [PV16,Theorem 4.5]. However, the number of minimal generators given in [PV16] is in general form and involves the rank of the zeroth homology groups of certain subcomplexes of the underlying classifying space.…”

Section: Introductionmentioning

confidence: 99%

“…In [PV16], Panov and Verëvkin constructed classifying spaces for the commutator subgroups of right-angled Coxeter groups and have given a general formula for the rank of such groups, see [PV16,Theorem 4.5]. However, the number of minimal generators given in [PV16] is in general form and involves the rank of the zeroth homology groups of certain subcomplexes of the underlying classifying space. As an immediate application of Theorem 1.1, we obtain the rank of T W n in terms of the 'arcs' of the twin group, or the 'dimension' of the cartographical group, and thus it is more explicit in our context.…”

Section: Introductionmentioning

confidence: 99%

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Commutator subgroups of twin groups and Grothendieck's cartographical groups

Dey

Gongopadhyay

2019

Journal of Algebra

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Let T Wn be the twin group on n arcs, n ≥ 2. The group T W m+2 is isomorphic to Grothendieck's m-dimensional cartographical group Cm, m ≥ 1. In this paper we give a finite presentation for the commutator subgroup T W m+2 , and prove that T W m+2 has rank 2m − 1. We derive that T W m+2 is free if and only if m ≤ 3. From this it follows that T W m+2 is word-hyperbolic and does not contain a surface group if and only if m ≤ 3. It also follows that the automorphism group of T W m+2 is finitely presented for m ≤ 3.As applications to the above results, we derive geometric properties of the ambient group T W m+2 . It is clear from the presentation in Theorem 1.1 that for m ≥ 4, T W m+2 contains free abelian subgroups of rank ≥ 2. By [Mou88, Theorem B], this shows that T W m+2 is not word-hyperbolic for m ≥ 4. Whereas from Corollary 1.4 we observe that T W m+2 is virtually free for m ≤ 3; so it is clear that T W m+2 is word-hyperbolic for m ≤ 3. Hence we have the following characterization for word-hyperbolicity of T W m+2 . Corollary 1.5. The group T W m+2 is word-hyperbolic if and only if m ≤ 3.Gordon, Long and Reid proved in [GLR04] that a coxeter group G is virtually free if and only if G does not contain a surface group. Since T W m+2 is finitely generated, by Corollary 1.5, it can not be virtually free for m ≥ 4. Hence we have the following.Corollary 1.6. The group T W m+2 does not contain a surface group if and only if m ≤ 3.

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“…Note that right-angled Coxeter groups are special instances of a more general construction called graph products of groups. The reader is referred to [22] for the study of the commutator subgroup of a general graph product of groups.…”

Section: Torsion In the Fundamental Groups Of Small Coversmentioning

confidence: 99%

Fundamental Groups of Small Covers Revisited

2019

International Mathematics Research Notices

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We study the topology of small covers from their fundamental groups. We find a way to obtain explicit presentations of the fundamental group of a small cover. Then we use these presentations to study the relations between the fundamental groups of a small cover and its facial submanifolds. In particular, we can determine when a facial submanifold of a small cover is π 1 -injective in terms of some purely combinatorial data on the underlying simple polytope. In addition, we find that any 3-dimensional small cover has an embedded non-simply-connected π 1 -injective surface. Using this result and some results of Schoen and Yau, we characterize all the 3-dimensional small covers that admit Riemannian metrics with nonnegative scalar curvature.

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Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

Cited by 32 publications

References 22 publications

Polyhedral products and features of their homotopy theory

Polyhedral products and features of their homotopy theory

Commutator subgroups of twin groups and Grothendieck's cartographical groups

Fundamental Groups of Small Covers Revisited

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