The geometric invariants of direct products of virtually free groups

Meinert, Holger

doi:10.1007/bf02564473

Cited by 16 publications

(20 citation statements)

References 15 publications

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“…By the convergence of the spectral sequence, H s ðS; QÞ is finite-dimensional if and only if E l i;sÀi is finite-dimensional for every i c s. Then, by (17), H s ðS; QÞ is finitedimensional if and only if E l 0;s is finite-dimensional. Now the last paragraph shows that H s ðS; QÞ is finite-dimensional if and only if E 2 0;s ¼ H s ðL; QÞ n Q½Q Q is finitedimensional.…”

Section: Subgroups With Finite-dimensional Homology Groupsmentioning

confidence: 98%

“…Motivated by the fact that the S-invariants determine the homological and homotopical types of subgroups above the commutator, Meinert [17] calculated the S-invariants of the direct product of finitely many virtually free groups. This calculation was later extended by Gehrke [12] to the direct product G 1 Â Á Á Á Â G m when S 1 ðG i Þ ¼ S l ðG i Þ and G i is FP l for every i.…”

Section: Corollary B Every Limit Group Is Free-by-(torsion-free Nilpmentioning

confidence: 99%

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On subdirect products of type FP m of limit groups

Kochloukova¹

2010

Journal of Group Theory

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Abstract. We show that limit groups are free-by-(torsion-free nilpotent) and have non-positive Euler characteristic. We prove that for any non-abelian limit group the Bieri-NeumannStrebel-Renz S-invariants are the empty set.Let s d 3 be a natural number and G be a subdirect product of non-abelian limit groups intersecting each factor non-trivially. We show that the homology groups of any subgroup of finite index in G, in dimension i c s and with coe‰cients in Q, are finite-dimensional if and only if the projection of G to the direct product of any s of the limit groups has finite index.

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Section: Subgroups With Finite-dimensional Homology Groupsmentioning

confidence: 98%

Section: Corollary B Every Limit Group Is Free-by-(torsion-free Nilpmentioning

confidence: 99%

On subdirect products of type FP m of limit groups

Kochloukova¹

2010

Journal of Group Theory

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“…Meinert [22] has calculated the Bieri-Neumann-Strebel invariants for direct products D of finitely many finitely generated free groups. In particular he has calculated which homomorphisms from D to an abelian group have a finitely presented kernel.…”

Section: Thus the Quotientsmentioning

confidence: 99%

Structure and finiteness properties of subdirect products of groups

Bridson

Miller

2008

Proceedings of the London Mathematical Society

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Abstract. We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup.

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“…First we construct the counterexamples. If G 1 and G 2 are both of type F n , then the following formula was conjectured in [19]:…”

Section: Some Examplesmentioning

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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The geometric invariants of direct products of virtually free groups

Cited by 16 publications

References 15 publications

On subdirect products of type FP m of limit groups

On subdirect products of type FP m of limit groups

Structure and finiteness properties of subdirect products of groups

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

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