The Dehn function of Baumslag’s metabelian group

Kassabov, Martin; Riley, Tim

doi:10.1007/s10711-011-9623-y

Cited by 11 publications

(18 citation statements)

References 24 publications

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“…Several of the techniques developed to deal with RAAGs have applications in other groups. "Pushing" may be used in the torsion analogs of Baumslag's metabelian group to find that the Dehn function is at worst quartic, as shown in [25]-a priori, it takes nontrivial work even to show that the Dehn function is polynomial. (In fact, it turns out to be quadratic, as shown by de Cornulier-Tessera in [10].)…”

Section: Introductionmentioning

confidence: 99%

Pushing fillings in right-angled Artin groups

Abrams

Brady

Dani

et al. 2012

Journal of the London Mathematical Society

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Abstract. We define a family of quasi-isometry invariants of groups called higher divergence functions, which measure isoperimetric properties "at infinity." We give sharp upper and lower bounds on the divergence functions for right-angled Artin groups, using different pushing maps on the associated cube complexes. In the process, we define a class of RAAGs we call orthoplex groups, which have the property that their Bestvina-Brady subgroups have hard-to-fill spheres. Our results give sharp bounds on the higher Dehn functions of Bestvina-Brady groups, a complete characterization of the divergence of geodesics in RAAGs, and an upper bound for filling loops at infinity in the mapping class group.

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Section: Introductionmentioning

confidence: 99%

Pushing fillings in right-angled Artin groups

Abrams

Brady

Dani

et al. 2012

Journal of the London Mathematical Society

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“…De Cornulier & Tessera showed that the Dehn function of Γ 2 (2) grows quadratically [18], and Kassabov & Riley [30] that that of Γ 2 grows exponentially.…”

Section: Cayley Graphs Of Generalized Lamplighter Groupsmentioning

confidence: 99%

Lamplighters, metabelian groups, and horocyclic products

Amchislavska¹,

Riley²

2016

Enseign. Math.

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Bartholdi, Neuhauser and Woess proved that a family of metabelian groups including lamplighters have a striking geometric manifestation as 1-skeleta of horocyclic products of trees. The purpose of this article is to give an elementary account of this result, to widen the family addressed to include the infinite valence case (for instance Z ≀ Z), and to make the translation between the algebraic and geometric descriptions explicit.In the rank-2 case, where the groups concerned include a celebrated example of Baumslag and Remeslennikov, we give the translation by means of a combinatorial 'lamplighter description'. This elucidates our proof in the general case which proceeds by manipulating polynomials.Additionally, we show that the Cayley 2-complex of a suitable presentation of Baumslag and Remeslennikov's example is a horocyclic product of three trees.

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“…It was shown in [3] that when p is prime, Γ 3 (p) is a cocompact lattice in Sol 5 (F p ((t))), and its Dehn function is quadratic. The Dehn function of Γ 3 (m) is studied for any m in [12] where it is shown to be at most quartic.…”

Section: Introductionmentioning

confidence: 99%