Graph algebras

Kim, Ki Hang; Makar-Limanov, Leonid; Neggers, J.; Roush, F.W.

doi:10.1016/0021-8693(80)90131-3

Cited by 42 publications

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“…The solution to the isomorphism problem has the strongest form: if G./ Š G. 0 / then Š 0 (see Droms [11] and Kim et al [20]). Servatius [26] conjectured a finite generating set for Aut.G.// and his conjecture was proved by Laurence [22].…”

Section: Introductionmentioning

confidence: 99%

The asymptotic geometry of right-angled Artin groups, I

2008

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We study atomic right-angled Artin groups -those whose defining graph has no cycles of length Ä 4, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.20F65, 20F69; 20F67, 05C25

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Section: Introductionmentioning

confidence: 99%

The asymptotic geometry of right-angled Artin groups, I

2008

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“…PT can be regarded as the presentation of a &-algebra kT, of a monoid Mr, or of a group Fr, called a graph group. These objects have been previously studied by various authors [2][3][4][5][6][7][8].…”

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confidence: 99%

Surface Subgroups of Graph Groups

Servatius¹,

Droms²,

Servatius³

1989

Proceedings of the American Mathematical Society

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Abstract.Given a graph T , define the group Fr to be that generated by the vertices of T, with a defining relation xy -yx for each pair x, y of adjacent vertices of T. In this article, we examine the groups Fr-, where the graph T is an H-gon, (n > 4). We use a covering space argument to prove that in this case, the commutator subgroup Ff contains the fundamental group of the orientable surface of genus 1 + (n -4)2"-3 . We then use this result to classify all finite graphs T for which Fp is a free group.To each graph T = ( V, E), with vertex set V and edge set E, we associate a presentation PT whose generators are the elements of V, and whose relations are {xy = yx\x ,y adjacent vertices of T).

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“…To a finite simple graph Γ = (V, E), there is associated a right-angled Artin group, G Γ , with a generator v for each vertex v ∈ V, and with a commutator relation vw = wv for each edge {v, w} ∈ E. The groups G Γ interpolate between the free groups F n (corresponding to the discrete graphs on n vertices, K n ), and the free abelian groups Z n (corresponding to the complete graphs on n vertices, K n ). These groups are completely determined by their underlying graphs: G Γ ∼ = G Γ ′ if and only if Γ ∼ = Γ ′ ; see [16,12]. In a previous paper [11], we settled Serre's question for the class of right-angled Artin groups: G Γ is quasi-projective iff G Γ is quasi-Kähler iff Γ is a complete multipartite graph (i.e., a join K n1,...,nr = K n1 * · · · * K nr of discrete graphs).…”

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confidence: 99%

Quasi-Kähler Bestvina-Brady groups

Dimca¹,

Papadima

Suciu

2008

J. Algebraic Geom.

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A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The Bestvina-Brady group N Γ is the kernel of the projection G Γ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kähler (respectively, Kähler) groups N Γ . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

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Graph algebras

Cited by 42 publications

References 0 publications

The asymptotic geometry of right-angled Artin groups, I

The asymptotic geometry of right-angled Artin groups, I

Surface Subgroups of Graph Groups

Quasi-Kähler Bestvina-Brady groups

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