Asymptotic dimension of finitely presented groups

Gentimis, Thanos

doi:10.1090/s0002-9939-08-08973-9

Cited by 28 publications

(28 citation statements)

References 11 publications

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“…This result without the VFP restriction for finitely presented groups was proven independently by Januszkiewicz-Swiatkowski [21], Gentimis [13], and Fujiwara-Whyte [12].…”

Section: Corollary 512 Every Group Of Type Vfp With Asdim ≤ 1 Is Virmentioning

confidence: 55%

Cohomological approach to asymptotic dimension

Dranishnikov

2008

Geom Dedicata

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We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X . In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe's coarse cohomology is strictly less than its asymptotic dimension.

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“…This result without the VFP restriction for finitely presented groups was proven independently by Januszkiewicz-Swiatkowski [21], Gentimis [13], and Fujiwara-Whyte [12].…”

Section: Corollary 512 Every Group Of Type Vfp With Asdim ≤ 1 Is Virmentioning

confidence: 55%

Cohomological approach to asymptotic dimension

Dranishnikov

2008

Geom Dedicata

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“…The following example is suggested by D Osin to us. We are informed by Dranishnikov that Corollary 1.2 has been known in work of Januszkiewicz and Swiatkowski [10] (see also Dranishnikov [4]) and that this example is also in Gentimis [7]. Example 1.3 Let G D A o ‫ޚ‬ be the wreath product such that A is a non-trivial finite group.…”

Section: Introductionmentioning

confidence: 99%

A note on spaces of asymptotic dimension one

Fujiwara

Whyte

2007

Algebr. Geom. Topol.

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“…In the case of asymptotic dimension 1, the construction above is optimal in the following sense. Januszkiewicz andŚwiatkowski [32] and independently Gentimis [22] proved that if a finitely presented group G has asymptotic dimension 1 then it is virtually free, and it follows that it satisfies τ 1,G n. So the groups Γ [19] all the groups considered in this section embed coarsely into a product of finitely many trees. It might be interesting to note that by arguments similar to those in Theorem 3.11, any such embedding must be strongly distorting, i.e.…”

Section: Applications To Dimension Theorymentioning

confidence: 82%

On exactness and isoperimetric profiles of discrete groups

Nowak

2007

Journal of Functional Analysis

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We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.A geometric characterization of amenable groups says that amenability is equivalent to existence of a sequence of Følner sets in the group. The degree of "how amenable" the group is can be measured by the growth rate of Følner sets for a fixed set of generators-this is the so called Følner function of an amenable group, it was introduced by Vershik in the 70's (see Section 1 for definitions).On the other hand one can study after Gromov the isoperimetric profile of any discrete group, i.e. a quantitative dependence between the volumes of a set A and its boundary ∂A. It is a consequence of Følner's characterization of amenability that in the case of amenable groups finding the asymptotics of the Følner function is equivalent to finding those of the isoperimetric profile. These functions also turned out to play a role in probability theory on groups after they were linked with random walks and the decay of the heat kernel by Varopoulos.

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Asymptotic dimension of finitely presented groups

Abstract: Abstract. We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than 1. It follows that a finitely presented group of asymptotic dimension 1 is virtually free.

Cited by 28 publications

References 11 publications

Cohomological approach to asymptotic dimension

Cohomological approach to asymptotic dimension

A note on spaces of asymptotic dimension one

On exactness and isoperimetric profiles of discrete groups

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