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The BNSR-invariants of the Houghton groups, concluded

Part of: Low-dimensional topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  15 July 2019

Matthew C. B. Zaremsky
Matthew C. B. Zaremsky*
Affiliation:
Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY12222, USA ([email protected])
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Abstract

We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.

Keywords

Houghton groupBNSR-invariantfiniteness propertiescube complexCAT(0)

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 57M07: Topological methods in group theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 1 , February 2020 , pp. 1 - 11
Copyright
Copyright © Edinburgh Mathematical Society 2019

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References

1.Bestvina, M. and Brady, N., Morse theory and finiteness properties of groups, Invent. Math. 129(3) (1997), 445–470.CrossRefGoogle Scholar
2.Bieri, R. and Renz, B., Valuations on free resolutions and higher geometric invariants of groups, Comment. Math. Helv. 63(3) (1988), 464–497.10.1007/BF02566775CrossRefGoogle Scholar
3.Bieri, R. and Sach, H., Groups of piecewise isometric permutations of lattice points (arXiv:1606.07728, 2016).Google Scholar
4.Bieri, R., Neumann, W. D. and Strebel, R., A geometric invariant of discrete groups, Invent. Math. 90(3) (1987), 451–477.CrossRefGoogle Scholar
5.Bieri, R., Geoghegan, R. and Kochloukova, D. H., The sigma invariants of Thompson's group F, Groups Geom. Dyn. 4(2) (2010), 263–273.CrossRefGoogle Scholar
6.Brown, K. S., Finiteness properties of groups, in Proceedings of the Northwestern conference on cohomology of groups (Evanston, IL, 1985), J. Pure Appl. Algebra 44 (1987), 45–75.10.1016/0022-4049(87)90015-6CrossRefGoogle Scholar
7.Bux, K.-U., Finiteness properties of soluble arithmetic groups over global function fields, Geom. Topol. 8 (2004), 611–644.CrossRefGoogle Scholar
8.Bux, K.-U. and Gonzalez, C., The Bestvina–Brady construction revisited: geometric computation of Σ-invariants for right-angled Artin groups, J. Lond. Math. Soc. (2) 60(3) (1999), 793–801.CrossRefGoogle Scholar
9.Haglund, F., Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209.CrossRefGoogle Scholar
10.Harlander, J. and Kochloukova, D. H., The Σ2-conjecture for metabelian groups: the general case, J. Algebra 273(2) (2004), 435–454.CrossRefGoogle Scholar
11.Houghton, C. H., The first cohomology of a group with permutation module coefficients, Arch. Math. (Basel) 31(3) (1978/79), 254–258.CrossRefGoogle Scholar
12.Kochloukova, D. H., Subgroups of constructible nilpotent-by-abelian groups and a generalization of a result of Bieri, Neumann and Strebel, J. Group Theory 5(2) (2002), 219–231.10.1515/jgth.5.2.219CrossRefGoogle Scholar
13.Kochloukova, D. H., On the Σ2-invariants of the generalised R. Thompson groups of type F, J. Algebra 371 (2012), 430–456.10.1016/j.jalgebra.2012.08.002CrossRefGoogle Scholar
14.Lee, S. R., Geometry of Houghton's groups (arXiv:1212.0257, 2012).Google Scholar
15.Meier, J., Meinert, H. and VanWyk, L., Higher generation subgroup sets and the Σ-invariants of graph groups, Comment. Math. Helv. 73(1) (1998), 22–44.10.1007/s000140050044CrossRefGoogle Scholar
16.Meinert, H., The homological invariants for metabelian groups of finite Prüfer rank: a proof of the Σm-conjecture, Proc. Lond. Math. Soc. (3) 72(2) (1996), 385–424.CrossRefGoogle Scholar
17.Sach, H., FPn properties of generalized Houghton groups (arXiv:1608.00933, 2016).Google Scholar
18.Witzel, S. and Zaremsky, M. C. B., The Σ-invariants of Thompson's group F via Morse theory, in Topological methods in group theory (eds. Broaddus, N., Davis, M., Lafont, J.-F. and Ortiz, I. J.), London Mathematical Society Lecture Note Series, Volume 451, pp. 173–193 (Cambridge University Press, Cambridge, 2018).Google Scholar
19.Witzel, S. and Zaremsky, M. C. B., The Basilica Thompson group is not finitely presented, Groups Geom. Dyn., in press.Google Scholar
20.Zaremsky, M. C. B., On the Σ-invariants of generalized Thompson groups and Houghton groups, Int. Math. Res. Not. IMRN 2017(19) (2017), 5861–5896.Google Scholar
21.Zaremsky, M. C. B., Geometric structures related to the braided Thompson groups (arXiv:1803.0271, 2018).Google Scholar