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.

Graphical splittings of Artin kernels

Journal of Group Theory ◽

10.1515/jgth-2020-0124 ◽

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 ◽

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.

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

10.1142/s0218196714500350 ◽

2014 ◽

Vol 24 (06) ◽

pp. 815-825 ◽

Cited By ~ 1

Author(s):

Matt Clay

Keyword(s):

Cyclic Subgroup ◽

Artin Group ◽

Artin Groups ◽

Cyclic Subgroups ◽

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.

Embeddability of right-angled Artin groups on complements of trees

10.1142/s0218196718500182 ◽

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 ◽

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.

SURFACE SUBGROUPS OF RIGHT-ANGLED ARTIN GROUPS

10.1142/s0218196708004536 ◽

2008 ◽

Vol 18 (03) ◽

pp. 443-491 ◽

Cited By ~ 11

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 ◽

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).

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

10.1142/s021819671450009x ◽

2014 ◽

Vol 24 (02) ◽

pp. 121-169 ◽

Cited By ~ 19

Author(s):

Sang-Hyun Kim ◽

Thomas Koberda

Keyword(s):

Mapping Class Groups ◽

Mapping Class ◽

Artin Group ◽

Artin Groups ◽

Class Groups ◽

Curve Graph ◽

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.

Right-angled Artin groups and enhanced Koszul properties

Journal of Group Theory ◽

10.1515/jgth-2020-0049 ◽

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 ◽

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.

ON THE RECOGNITION OF RIGHT-ANGLED ARTIN GROUPS

Glasgow Mathematical Journal ◽

10.1017/s0017089519000235 ◽

2019 ◽

Vol 62 (2) ◽

pp. 473-475

Author(s):

MARTIN R. BRIDSON

Keyword(s):

Artin Group ◽

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.

Right-angled Artin groups and full subgraphs of graphs

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500596 ◽

2017 ◽

Vol 26 (10) ◽

pp. 1750059 ◽

Cited By ~ 1

Author(s):

Takuya Katayama

Keyword(s):

Finite Graph ◽

Artin Group ◽

Artin Groups ◽

Linear Forest ◽

Complement Graph ◽

The Right

For a finite graph [Formula: see text], let [Formula: see text] be the right-angled Artin group defined by the complement graph of [Formula: see text]. We show that, for any linear forest [Formula: see text] and any finite graph [Formula: see text], [Formula: see text] can be embedded into [Formula: see text] if and only if [Formula: see text] can be realized as a full subgraph of [Formula: see text]. We also prove that if we drop the assumption that [Formula: see text] is a linear forest, then the above assertion does not hold, namely, for any finite graph [Formula: see text], which is not a linear forest, there exists a finite graph [Formula: see text] such that [Formula: see text] can be embedded into [Formula: see text], though [Formula: see text] cannot be embedded into [Formula: see text] as a full subgraph.

Quasi-isometries

Office Hours with a Geometric Group Theorist ◽

10.23943/princeton/9780691158662.003.0007 ◽

2017 ◽

Author(s):

Dan Margalit ◽

Anne Thomas

Keyword(s):

Group Theory ◽

Algebraic Structure ◽

Cayley Graphs ◽

Geometric Group Theory ◽

Schwarz Lemma ◽

Research Projects ◽

Geometric Properties ◽

Fundamental Lemma ◽

Lipschitz Equivalence ◽

Sample Result

This chapter considers the notion of quasi-isometry, also known as “coarse isometry.” A whole suite of important algebraic and geometric properties is preserved by quasi-isometries. Quasi-isometry can be applied to the algebraic structure of groups. A sample result, which shows that quasi-isometries can have powerful algebraic consequences, is a theorem of Gromov. Along the way to this theorem, the chapter proves the Milnor–Schwarz lemma, sometimes referred to as the fundamental lemma of geometric group theory. After describing Cayley graphs as well as path metrics and word metrics for integers, the chapter explores the bi-Lipschitz equivalence of word metrics, quasi-isometric equivalence of Cayley graphs, quasi-isometries between groups and spaces, and quasi-isometric rigidity. The discussion includes exercises and research projects.

The Ping-Pong Lemma

Office Hours with a Geometric Group Theorist ◽

10.23943/princeton/9780691158662.003.0005 ◽

2017 ◽

Author(s):

Johanna Mangahas

Keyword(s):

Group Theory ◽

Free Group ◽

Group Action ◽

Metric Space ◽

Hyperbolic Geometry ◽

Free Groups ◽

Central Place ◽

Geometric Group Theory ◽

Research Projects ◽

Ping Pong

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.