scholarly journals On subgroups of right angled Artin groups with few generators

2015 ◽  
Vol 25 (04) ◽  
pp. 675-688 ◽  
Author(s):  
Ashot Minasyan
Keyword(s):  
Direct Product ◽  
Free Group ◽  
Free Groups ◽  
Cohomological Dimension ◽  
Artin Group ◽  
Direct Products ◽  
Artin Groups ◽  
Limit Groups ◽  
Right Angled Artin Groups ◽  
Finitely Presented

For each d ∈ ℕ, we construct a 3-generated group Hd, which is a subdirect product of free groups, such that the cohomological dimension of Hd is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group Gd, embeddable into a right angled Artin group, such that the cohomological dimension of Gd is 2 but the cohomological dimension of any right angled Artin group, containing Gd, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit 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.

scholarly journals TOPOLOGY OF RANDOM RIGHT ANGLED ARTIN GROUPS

2011 ◽  
Vol 03 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ARMINDO COSTA ◽  
MICHAEL FARBER
Keyword(s):  
Motion Planning ◽  
Random Graphs ◽  
Betti Numbers ◽  
Cohomological Dimension ◽  
Topological Invariants ◽  
Topological Complexity ◽  
Artin Group ◽  
Artin Groups ◽  
Right Angled Artin Groups ◽  
Random Groups

In this paper, we study topological invariants of a class of random groups. Namely, we study right angled Artin groups associated to random graphs and investigate their Betti numbers, cohomological dimension and topological complexity. The latter is a numerical homotopy invariant reflecting complexity of motion planning algorithms in robotics. We show that the topological complexity of a random right angled Artin group assumes, with probability tending to one, at most three values, when n → ∞. We use a result of Cohen and Pruidze which expresses the topological complexity of right angled Artin groups in combinatorial terms. Our proof deals with the existence of bi-cliques in random graphs.

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 On commuting elements and embeddings of graph groups and monoids

2009 ◽  
Vol 52 (1) ◽  
pp. 155-170 ◽  
Author(s):  
Mark Kambites
Keyword(s):  
Direct Product ◽  
Abelian Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups ◽  
Graph Groups

AbstractWe study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

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