scholarly journals Poly-freeness of Artin groups and the Farrell–Jones Conjecture

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiaolei Wu
Keyword(s):  
Artin Group ◽  
Artin Groups

Abstract We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let Γ be a finite simplicial graph with all edges labelled by positive even integers, and let A Γ A_{\Gamma} be its associated Artin group; our new proof implies that if A T A_{T} is poly-free (resp. normally poly-free) for every clique 𝑇 in Γ, then A Γ A_{\Gamma} is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell–Jones Conjecture for even Artin groups. In particular, we show that if A Γ A_{\Gamma} is an even Artin group such that each clique in Γ either has at most three vertices, has all of its labels at least 6, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by 2), then A Γ A_{\Gamma} satisfies the Farrell–Jones Conjecture. In addition, our methods enable us to obtain results for general Artin groups.

scholarly journals Graphical splittings of Artin kernels

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrique Miguel Barquinero ◽  
Lorenzo Ruffoni ◽  
Kaidi Ye
Keyword(s):  
Free Group ◽  
Artin Group ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Block Graphs ◽  
Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

When does a right-angled Artin group split over ℤ?

2014 ◽  
Vol 24 (06) ◽  
pp. 815-825 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Cyclic Subgroup ◽  
Artin Group ◽  
Artin Groups ◽  
Cyclic Subgroups ◽  
Right Angled Artin Groups ◽  
Jsj Decompositions

We show that a right-angled Artin group, defined by a graph Γ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if Γ is biconnected. Further, we compute JSJ-decompositions of 1-ended right-angled Artin groups over infinite cyclic subgroups.

scholarly journals Embeddability of right-angled Artin groups on complements of trees

2018 ◽  
Vol 28 (03) ◽  
pp. 381-394
Author(s):  
Eon-Kyung Lee ◽  
Sang-Jin Lee
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Graph Theorem ◽  
Induced Subgraph ◽  
Path Graph ◽  
Right Angled Artin Groups ◽  
The Right

For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is the complement of a tree and for the case [Formula: see text] is the complement of a path graph.

scholarly journals SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS

2008 ◽  
Vol 18 (03) ◽  
pp. 443-491 ◽  
Author(s):  
JOHN CRISP ◽  
MICHAH SAGEEV ◽  
MARK SAPIR
Keyword(s):  
Sufficient Conditions ◽  
First Integral ◽  
Closed Surface ◽  
Hyperbolic Surface ◽  
Artin Group ◽  
Artin Groups ◽  
Diagram Groups ◽  
Group A ◽  
Right Angled Artin Groups ◽  
Diagram Group

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight "forbidden" graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right-angled Artin group to contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs P2(6), the right-angled Artin group A(P2(6)) is a subgroup of a (right-angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

scholarly journals The geometry of the curve graph of a right-angled Artin group

2014 ◽  
Vol 24 (02) ◽  
pp. 121-169 ◽  
Author(s):  
Sang-Hyun Kim ◽  
Thomas Koberda
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Artin Group ◽  
Artin Groups ◽  
Class Groups ◽  
Curve Graph ◽  
Right Angled Artin Groups ◽  
Central Result ◽  
The Right ◽  
Image Theorem

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

scholarly journals Right-angled Artin groups and enhanced Koszul properties

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alberto Cassella ◽  
Claudio Quadrelli
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Galois Cohomology ◽  
Finite Graph ◽  
Galois Groups ◽  
Cohomology Algebra ◽  
Artin Group ◽  
Artin Groups ◽  
Koszul Algebra ◽  
Right Angled Artin Groups

AbstractLet 𝔽 be a finite field. We prove that the cohomology algebra H^{\bullet}(G_{\Gamma},\mathbb{F}) with coefficients in 𝔽 of a right-angled Artin group G_{\Gamma} is a strongly Koszul algebra for every finite graph Γ. Moreover, H^{\bullet}(G_{\Gamma},\mathbb{F}) is a universally Koszul algebra if, and only if, the graph Γ associated to the group G_{\Gamma} has the diagonal property. From this, we obtain several new examples of pro-𝑝 groups, for a prime number 𝑝, whose continuous cochain cohomology algebra with coefficients in the field of 𝑝 elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to a conjecture on Galois cohomology of maximal pro-𝑝 Galois groups of fields formulated by J. Mináč et al.

Right-Angled Artin Groups

2017 ◽  
Author(s):  
Robert W. Bell ◽  
Matt Clay
Keyword(s):  
Group Theory ◽  
Broad Spectrum ◽  
Geometric Group Theory ◽  
The Other ◽  
Artin Group ◽  
Artin Groups ◽  
Research Projects ◽  
Open Questions ◽  
Free Abelian Groups ◽  
Right Angled Artin Groups

This chapter deals with right-angled Artin groups, a broad spectrum of groups that includes free groups on one end, free abelian groups on the other end, and many other interesting groups in between. A right-angled Artin group is a group G(Γ‎) defined in terms of a graph Γ‎. Right-angled Artin groups have taken a central role in geometric group theory, mainly due to their involvement in the solution to one of the main open questions in the topology of 3-manifolds. The chapter first considers right-angled Artin groups as subgroups and how they relate to other classes of groups before exploring subgroups of right-angled Artin groups and the word problem for right-angled Artin groups. The discussion includes exercises and research projects.

scholarly journals ON THE RECOGNITION OF RIGHT-ANGLED ARTIN GROUPS

2019 ◽  
Vol 62 (2) ◽  
pp. 473-475
Author(s):  
MARTIN R. BRIDSON
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Right Angled Artin Groups

AbstractThere does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group.

scholarly journals BUSEMANN POINTS OF ARTIN GROUPS OF DIHEDRAL TYPE

2009 ◽  
Vol 19 (07) ◽  
pp. 891-910
Author(s):  
CORMAC WALSH
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Growth Series ◽  
Boundary Points ◽  
Geodesic Rays ◽  
Dihedral Type

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is, the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterize the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.

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