The Homological Invariants for Metabelian Groups of Finite Prüfer Rank: A Proof of the Σ<sup>m</sup>
-Conjecture

Meinert, Holger

doi:10.1112/plms/s3-72.2.385

Cited by 30 publications

(22 citation statements)

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“…The homological case of (1) for all m is [24], Proposition 4.2, and the homotopical case for m D 2 is a special case of [25], Theorem 4.3. The homotopical case of (1) for all m then follows.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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The Sigma invariants of Thompson’s group $F$

Bieri¹,

Geoghegan²,

Kochloukova³

2010

Groups Geom. Dyn.

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Abstract. Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show thatAs an application, we show that, for every m, F has subgroups of type F m 1 which are not of type FP m (thus certainly not of type F m ).Mathematics Subject Classification (2010). 20J05, 20F65, 55U10.

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Section: Proof Of Theoremmentioning

confidence: 99%

“…† m .GI R/) classifies all normal subgroups N of G containing the commutator subgroup G 0 by their finiteness properties in the following sense: Theorem 1.1 ([7], [27], [28] [24]. A complete description of † m .G/ and † m .GI Z/ for any right angled Artin group G is given in [23].…”

Section: Some Facts About Sigma Invariantsmentioning

confidence: 99%

The Sigma invariants of Thompson’s group $F$

Bieri¹,

Geoghegan²,

Kochloukova³

2010

Groups Geom. Dyn.

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“…The homological version of this Theorem was originally obtained by Schmitt [24]. Proofs can be found in [21] for the homological case, and [22] in the homotopical case.…”

Section: Ii) If Either the Trivial Group Does Not Belong To H Or The mentioning

confidence: 99%

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

Meier¹,

Meinert²,

VanWyk³

1998

Comment. Math. Helv.

118

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Abstract. We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character χ ∈ Hom(G, ) represents a point in the homotopical or homological Σ-invariants of the group G. Let G be a finite simplicial graph, G the flag complex induced by G, and GG the graph group, or 'right angled Artin group', defined by G. We use our result on n-generating subgroup sets to describe the homotopical and homological Σ-invariants of GG in terms of the topology of subcomplexes of G. In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the Σ-invariants for a family of groups whose higher invariants are not determined -either implicitly or explicitly -by Σ 1 . Mathematics Subject Classification (1991). 57M07, 20F36.

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“…ZG and we obtain a long exact sequence (see Brown [6,Chapter VII.9 We note the following immediate corollary which is well known; see Meier, Meinert and VanWyk [19] and Meinert [20]. Corollary 2.6 Let W G !…”

Section: Sigma Invariants and Novikov Ringsmentioning

confidence: 99%

Novikov homology of HNN–extensions and right-angled Artin groups

Schütz

2009

Algebr. Geom. Topol.

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We calculate the Novikov homology of right-angled Artin groups and certain HNNextensions of these groups. This is used to obtain information on the homological Sigma invariants of Bieri-Neumann-Strebel-Renz for these groups. These invariants are subsets of all homomorphisms from a group to the reals containing information on the finiteness properties of kernels of such homomorphisms. We also derive information on the homotopical Sigma invariants and show that one cannot expect any symmetry relations between a homomorphism and its negative regarding these invariants. While it was previously known that these invariants are not symmetric in general, we give the first examples of homomorphisms which are symmetric with respect to the homological invariant, but not with respect to the homotopical invariant. 20J05; 20F65, 57R19

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The Homological Invariants for Metabelian Groups of Finite Prüfer Rank: A Proof of the Σ^m -Conjecture

Cited by 30 publications

References 0 publications

The Sigma invariants of Thompson’s group $F$

The Sigma invariants of Thompson’s group $F$

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

Novikov homology of HNN–extensions and right-angled Artin groups

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The Homological Invariants for Metabelian Groups of Finite Prüfer Rank: A Proof of the Σm -Conjecture

Cited by 30 publications

References 0 publications

The Sigma invariants of Thompson’s group $F$

The Sigma invariants of Thompson’s group $F$

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

Novikov homology of HNN–extensions and right-angled Artin groups

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Product

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The Homological Invariants for Metabelian Groups of Finite Prüfer Rank: A Proof of the Σ^m -Conjecture