Connectivity properties of group actions on non-positively curved spaces

Bieri, Robert; Geoghegan, Ross

doi:10.1090/memo/0765

Cited by 30 publications

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“…The definition of † 1 given here agrees with the now-established conventions followed, for example, in [7] and in [2]. It differs by a sign from the † 1 -invariant defined in [6].…”

Section: The Homological Casesupporting

confidence: 69%

“…A complete description of † m .G/ and † m .GI Z/ for any right angled Artin group G is given in [23]. Recently the homotopical invariant † m .G/ has been generalized to an invariant of group actions on proper CAT(0) metric spaces [2]; the corresponding invariants for the natural action of SL n .R/ on its symmetric space have been calculated: for n D 2 (action by Möbius transformations on the hyperbolic plane) in [3], and for n > 2 in [26]. A similar generalization of the homological case, † m .GI R/, to the CAT(0) setting will appear in [5].…”

Section: Some Facts About Sigma Invariantsmentioning

confidence: 99%

“…1 Here, PL homeomorphisms are understood to act on OE0; 1 on the left as in [15] rather than on the right as in [13]. 2 See Subsection 2.1 for the definition.…”

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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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“…The definition of † 1 given here agrees with the now-established conventions followed, for example, in [7] and in [2]. It differs by a sign from the † 1 -invariant defined in [6].…”

Section: The Homological Casesupporting

confidence: 69%

Section: Some Facts About Sigma Invariantsmentioning

confidence: 99%

See 1 more Smart Citation

The Sigma invariants of Thompson’s group $F$

Bieri¹,

Geoghegan²,

Kochloukova³

2010

Groups Geom. Dyn.

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“…S m ðGÞ and S m ðG; ZÞ) in arbitrary dimension m d 1; see [5], [6], [19]. These invariants are subsets of the character sphere Recently the homotopical invariant was generalized to the case of groups acting on a CAT(0)-space [4]. The importance of S invariants is that they capture enough information to deduce the types FP s and F s of a subgroup that contains the commutator [6].…”

Section: Conjecture Letmentioning

confidence: 99%

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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“…This topological approach has become more important in recent years. Bieri and Geoghegan [2] extend this theory to isometry actions of a group G on a CAT(0) space M . Although we will restrict ourselves to the classical case, we will use this more modern approach for our definitions.…”

Section: Introductionmentioning

confidence: 95%

Controlled connectivity of closed 1–forms

Schütz

2002

Algebr. Geom. Topol.

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We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel-Renz invariant. It is also related to the Morse theory of closed 1-forms. Given a controlled 0-connected cohomology class on a manifold M with n = dim M ≥ 5 we can realize it by a closed 1-form which is Morse without critical points of index 0, 1, n − 1 and n. If n = dim M ≥ 6 and the cohomology class is controlled 1-connected we can approximately realize any chain complex D * with the simple homotopy type of the Novikov complex and with D i = 0 for i ≤ 1 and i ≥ n − 1 as the Novikov complex of a closed 1-form. This reduces the problem of finding a closed 1-form with a minimal number of critical points to a purely algebraic problem.

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Connectivity properties of group actions on non-positively curved spaces

Abstract: ScienceFoundation.1 G has type F n if there is a K(G, 1)-complex with finite n-skeleton. All groups have type F 0 , F 1 is "finitely generated", F 2 is "finitely presented", etc.

Cited by 30 publications

References 0 publications

The Sigma invariants of Thompson’s group $F$

The Sigma invariants of Thompson’s group $F$

On subdirect products of type FP m of limit groups

Controlled connectivity of closed 1–forms

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