Groups not presentable by products

Kotschick, D.; Löh, Clara

doi:10.4171/ggd/180

Cited by 7 publications

(17 citation statements)

References 43 publications

(98 reference statements)

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“…The following list of problems from [17, Section 1] and [4, Section 9: Appendix II] (see also [8,Section 5.35]) is the main motivation for this paper: The assumptions of Theorem 1.4 occur naturally quite often. On the one hand, there are plenty of examples of manifolds that do not admit maps of non-zero degree from direct products [9,10,12,16,15]. On the other hand, the assumption that M cannot be realized by a cohomology class in H m (N; Q) is fulfilled in several instances, e.g.…”

Section: Example 12 Originates From the Following General Questionmentioning

confidence: 99%

Degrees of Self-Maps of Products

Neofytidis

2016

Int Math Res Notices

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Every closed oriented manifold M is associated with a set of integers D(M ), the set of self-mapping degrees of M . In this paper we investigate whether a product M × N admits a self-map of degree d, when neither D(M ) nor D(N ) contains d. We find sufficient conditions so that D(M × N ) contains exactly the products of the elements of D(M ) with the elements of D(N ). As a consequence, we obtain manifolds M × N that do not admit self-maps of degree −1 (strongly chiral), that have finite sets of self-mapping degrees (inflexible) and that do not admit any self-map of degree dp for a prime number p. Furthermore we obtain a characterization of odd-dimensional strongly chiral hyperbolic manifolds in terms of self-mapping degrees of their products.

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Section: Example 12 Originates From the Following General Questionmentioning

confidence: 99%

Degrees of Self-Maps of Products

Neofytidis

2016

Int Math Res Notices

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“…Compact manifolds with the latter geometry are finitely covered by products, whereas those with the former geometry are not even dominated by products, although the fundamental groups are presentable by products in both cases. It was noted in [15,Thm. 10.2] that presentability by products is not a quasi-isometry invariant property of finitely generated groups.…”

Section: Three-manifold Groups Presentable By Productsmentioning

confidence: 99%

“…This property was introduced in [14] and further studied in [15] because, according to [14], it is a property that the fundamental groups of rationally essential manifolds dominated by products must have. Proof.…”

Section: Three-manifold Groups Presentable By Productsmentioning

confidence: 99%

“…By [15,Cor. 9.2] the only non-trivial free product that is presentable by a product is Z 2 ⋆ Z 2 , which is virtually Z and so satisfies (2).…”

Section: Three-manifold Groups Presentable By Productsmentioning

confidence: 99%

“…This is motivated by the work of Löh and the first author in [14,15], where strong restrictions were found for certain manifolds with large universal coverings to be dominated by products. In fact, the results of those papers show that three-manifolds dominated by products cannot have hyperbolic or Sol 3 -geometry, and must often be prime.…”

Section: Introductionmentioning

confidence: 99%

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On three-manifolds dominated by circle bundles

Kotschick

Neofytidis

2012

Math. Z.

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We determine which three-manifolds are dominated by products. The result is that a closed, oriented, connected three-manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of S 2 × S 1 . This characterization can also be formulated in terms of Thurston geometries, or in terms of purely algebraic properties of the fundamental group. We also determine which three-manifolds are dominated by non-trivial circle bundles, and which three-manifold groups are presentable by products.

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