Right-Angled Artin Groups

A right-angled Artin group (often abbrivated as RAAG) is a group which has a presentation whose only relations are commutators between generators. These are also known as graph groups, since they are (uniquely) encoded by (simple) graphs, or partially commmutative groups.

AUTHORS:

  • Travis Scrimshaw (2013-09-01): Initial version
class sage.groups.raag.RightAngledArtinGroup(G)

Bases: sage.groups.finitely_presented.FinitelyPresentedGroup

The right-angled Artin group defined by a graph \(G\).

Let \(\Gamma = \{V(\Gamma), E(\Gamma)\}\) be a simple graph. A right-angled Artin group (commonly abbriated as RAAG) is the group

\[A_{\Gamma} = \langle g_v : v \in V(\Gamma) \mid [g_u, g_v] \text{ if } \{u, v\} \notin E(\Gamma) \rangle.\]

These are sometimes known as graph groups or partitally commutative groups. This RAAG’s contains both free groups, given by the complete graphs, and free abelian groups, given by disjoint vertices.

Warning

This is the opposite convention of some papers.

Right-angled Artin groups contain many remarkable properties and have a very rich structure despite their simple presentation. Here are some known facts:

  • The word problem is solvable.
  • They are known to be rigid; that is for any finite simple graphs \(\Delta\) and \(\Gamma\), we have \(A_{\Delta} \cong A_{\Gamma}\) if and only if \(\Delta \cong \Gamma\) [Droms1987].
  • They embed as a finite index subgroup of a right-angled Coxeter group (which is the same definition as above except with the additional relations \(g_v^2 = 1\) for all \(v \in V(\Gamma)\)).
  • In [BB1997], it was shown they contain subgroups that statisfy the property \(FP_2\) but are not finitely presented by considering the kernal of \(\phi : A_{\Gamma} \to \ZZ\) by \(g_v \mapsto 1\) (i.e. words of exponent sum 0).
  • \(A_{\Gamma}\) has a finite \(K(\pi, 1)\) space.
  • \(A_{\Gamma}\) acts freely and cocompactly on a finite dimensional \(CAT(0)\) space, and so it is biautomatic.
  • Given an Artin group \(B\) with generators \(s_i\), then any subgroup generated by a collection of \(v_i = s_i^{k_i}\) where \(k_i \geq 2\) is a RAAG where \([v_i, v_j] = 1\) if and only if \([s_i, s_j] = 1\) [CP2001].

The normal forms for RAAG’s in Sage are those described in [VW1994] and gathers commuting groups together.

EXAMPLES:

sage: Gamma = Graph(4)
sage: G = RightAngledArtinGroup(Gamma)
sage: a,b,c,d = G.gens()
sage: a*c*d^4*a^-3*b
v0^-2*v1*v2*v3^4

sage: Gamma = graphs.CompleteGraph(4)
sage: G = RightAngledArtinGroup(Gamma)
sage: a,b,c,d = G.gens()
sage: a*c*d^4*a^-3*b
v0*v2*v3^4*v0^-3*v1

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G
Right-angled Artin group of Cycle graph
sage: a,b,c,d,e = G.gens()
sage: e^-1*c*b*e*b^-1*c^-4
v2^-3

REFERENCES:

[Charney2006]Ruth Charney. An introduction to right-angled Artin groups. http://people.brandeis.edu/~charney/papers/RAAGfinal.pdf, Arxiv math/0610668.
[BB1997]Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups. Invent. Math. 129 (1997). No. 3, 445-470. www.math.ou.edu/~nbrady/papers/morse.ps.
[Droms1987]Carl Droms. Isomorphisms of graph groups. Proc. of the Amer. Math. Soc. 100 (1987). No 3. http://educ.jmu.edu/~dromscg/vita/preprints/Isomorphisms.pdf
[CP2001]John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math. 145 (2001). No 1, 19-36. Arxiv math/0003133.
[VW1994]Leonard Van Wyk. Graph groups are biautomatic. J. Pure Appl. Alg. 94 (1994). no. 3, 341-352.
class Element(parent, lst)

Bases: sage.groups.finitely_presented.FinitelyPresentedGroupElement

An element of a right-angled Artin group (RAAG).

Elements of RAAGs are modeled as lists of pairs [i, p] where i is the index of a vertex in the defining graph (with some fixed order of the vertices) and p is the power.

RightAngledArtinGroup.as_permutation_group()

Raise a ValueError error since right-angled Artin groups are infinite, so they have no isomorphic permutation group.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.as_permutation_group()
Traceback (most recent call last):
...
ValueError: the group is infinite
RightAngledArtinGroup.cardinality()

Return the number of group elements.

OUTPUT:

Infinity.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.cardinality()
+Infinity
RightAngledArtinGroup.gen(i)

Return the i-th generator of self.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.gen(2)
v2
RightAngledArtinGroup.gens()

Return the generators of self.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.gens()
(v0, v1, v2, v3, v4)
sage: Gamma = Graph([('x', 'y'), ('y', 'zeta')])
sage: G = RightAngledArtinGroup(Gamma)
sage: G.gens()
(vx, vy, vzeta)
RightAngledArtinGroup.graph()

Return the defining graph of self.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.graph()
Cycle graph: Graph on 5 vertices
RightAngledArtinGroup.ngens()

Return the number of generators of self.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.ngens()
5
RightAngledArtinGroup.one()

Return the identity element \(1\).

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.one()
1
RightAngledArtinGroup.one_element()

Return the identity element \(1\).

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.one()
1
RightAngledArtinGroup.order()

Return the number of group elements.

OUTPUT:

Infinity.

EXAMPLES:

sage: Gamma = graphs.CycleGraph(5)
sage: G = RightAngledArtinGroup(Gamma)
sage: G.cardinality()
+Infinity

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