Topology of Random Right Angled Artin Groups

Costa, A.; Farber, Michael

doi:10.1142/s1793525311000490

Cited by 7 publications

(16 citation statements)

References 31 publications

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“…Since right-angled Artin groups are indexed over graphs, it is natural to ask about the properties of random ones. Random right-angled Artin groups were studied by Costa-Farber in [3], and their automorphism groups were specifically studied by Charney-Farber in [2]. Charney and Farber showed that under certain conditions, a random right-angled Artin group almost certainly has a finite outer automorphism group; the results of this paper are a sharpening of their results.…”

Section: Introductionmentioning

confidence: 67%

Finiteness of outer automorphism groups of random right-angled Artin groups

Day

2012

Algebr. Geom. Topol.

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We consider the outer automorphism group Out.A / of the right-angled Artin group A of a random graph on n vertices in the Erdős-Rényi model. We show that the functions n 1 .log.n/ C log.log.n/// and 1 n 1 .log.n/ C log.log.n/// bound the range of edge probability functions for which Out.A / is finite: if the probability of an edge in is strictly between these functions as n grows, then asymptotically Out.A / is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out.A / is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.05C80, 20E36, 20F28, 20F69; 20F05

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Section: Introductionmentioning

confidence: 67%

Finiteness of outer automorphism groups of random right-angled Artin groups

Day

2012

Algebr. Geom. Topol.

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“…if 1 p.n/ D log n !.n/ n . In [12], Costa and the second author analyze the cohomological dimension and the topological complexity of right-angled Artin groups associated to random graphs.…”

Section: The Automorphism Groups Of Random Right-angled Artin Groupsmentioning

confidence: 99%

“…Random right-angled Artin groups, which were first studied in Costa-Farber [12], represent a different class of random groups. The right-angled Artin group associated to a graph is the group generated by the vertex set of with commutation relations between adjacent vertices.…”

Section: Introductionmentioning

confidence: 99%

Random groups arising as graph products

Charney

Farber

2012

Algebr. Geom. Topol.

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In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdős -Rényi model of a random graph and find precise threshold functions for the hyperbolicity (or relative hyperbolicity). We aslo study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as n → ∞, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter p is constant and satisfies 0.2929 < p < 1.

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“…The groups G(n, p, Γ) were considered previously by Charney-Farber [7]. Somewhat earlier, Costa-Farber [8] had looked at the special case of the random right-angled Artin group A G(n,p) . A formula for the cohomological dimension of A G(n,p) (= 1 + dim X(n, p)) in terms of (n, p) can be found in [8], as well as, a formula for the 'topological complexity' of its classifying space.…”

Section: Introductionmentioning

confidence: 99%

“…Somewhat earlier, Costa-Farber [8] had looked at the special case of the random right-angled Artin group A G(n,p) . A formula for the cohomological dimension of A G(n,p) (= 1 + dim X(n, p)) in terms of (n, p) can be found in [8], as well as, a formula for the 'topological complexity' of its classifying space. It is noted in [7] that if each Γ i is finite, then the graph product G(n, p, Γ) is word hyperbolic if and only if G(n, p) has no empty (induced) 4-cycles; furthermore, it is determined when this condition holds 'with high probability' (w.h.p.…”

Section: Introductionmentioning

confidence: 99%

Random graph products of finite groups are rational duality groups

Davis

Kahle

2014

Journal of Topology

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Given an edge-independent random graph G(n, p), we determine various facts about the cohomology of graph products of groups for the graph G(n, p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n → ∞. This includes random right-angled Coxeter groups as a special case. . convention, the empty set is considered a simplex in any simplicial complex. Given σ ∈ X, its link, denoted by Lk(σ, X) (or sometimes simply Lk(σ)), is the simplicial complex whose poset of nonempty simplices is isomorphic to X >σ (:= {τ ∈ X | τ > σ}).Let P(I) denote the power set of a finite set I. Given an I-tuple t = (t i ) i∈I of indeterminates and J ∈ P(I), define a monomial t J by t J = j∈J t j .(2.1)The f -polynomial of X is the polynomial in t = (t i ) i∈ [n] defined by f X (t) := σ∈X t σ .

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Topology of Random Right Angled Artin Groups

Cited by 7 publications

References 31 publications

Finiteness of outer automorphism groups of random right-angled Artin groups

Finiteness of outer automorphism groups of random right-angled Artin groups

Random groups arising as graph products

Random graph products of finite groups are rational duality groups

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