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Tubular Free by Cyclic Groups Act Freely on CAT(0) Cube Complexes

Published online by Cambridge University Press:  20 November 2018

Jack Button
Jack Button*
Affiliation:
Selwyn College, University of Cambridge, Cambridge CB3 9DQ, UK e-mail: j.o.button@dpmms.cam.ac.uk
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Abstract

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We identify when a tubular group (the fundamental group of a finite graph of groups with ${{\mathbb{Z}}^{2}}$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise’s equitable sets criterion, that every tubular free by cyclic group acts freely on a $\text{CAT}\left( 0 \right)$ cube complex.

Keywords

20F6520F6720E08CAT(0)tubular group
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 54 - 62
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Agol, I., The virtual Haken conjecture. Doc. Math. 18(2013), 1045–1087.Google Scholar
[2] Bestvina, M. and Feighn, M., A combination theorem for negatively curved groups. J. Differential Geom. 35(1992), 85–101.Google Scholar
[3] Brady, N. and Bridson, M. R., There is only one gap in the isoperimetric spectrum. Geom. Funct. Anal. 10(2000), 1053–1070. http://dx.doi.Org/10.1007/PL00001 646 Google Scholar
[4] Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. http://dx.doi.Org/10.1007/978-3-662-12494-9 Google Scholar
[5] Brinkmann, P. Hyperbolic automorphisms of free groups. Geom. Funct. Anal. 10(2000), no. 5, 1071–1089. http://dx.doi.Org/10.1007/PL00001 647 Google Scholar
[6] Burger, M. and Mozes, S., Finitely presented simple groups and products of trees. C. R. Acad. Sci. Paris Ser. I Math. 324(1997), no. 7, 747–752. http://dx.doi.Org/10.101 6/S0764-4442(97)86938-8 Google Scholar
[7] Button, J. O., Large groups of deficiency 1. Israel J. Math. 167(2008) 111–140. http://dx.doi.Org/10.1007/s11856-008-1043-9 Google Scholar
[8] Cashen, C. H., Quasi-isometries between tubular groups. Groups Geom. Dyn. 4(2010), no. 3, 473–516. http://dx.doi.Org/10.4171/CCD/92 Google Scholar
[9] Cashen, C. H. and Levitt, G., Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups. J. Group Theory 19(2016), no. 2,191-216. http://dx.doi.Org/10.151S/jgth-2015-0038 Google Scholar
[10] Gersten, S. M., The automorphism group of a free group is not a CAT(0) group. Proc. Amer. Math. Soc. 121(1994), no. 4, 999–1002. http://dx.doi.Org/10.2307/2161207 Google Scholar
[11] Hagen, M. F. and Wise, D. T., Cubulating hyperbolic free-by-cydic groups: the general case. Geom. Funct. Anal. 25(2015), no. 1, 134–179. http://dx.doi.Org/10.1007/s00039-015-0314-y Google Scholar
[12] Haglund, F. and Wise, D. T., Special cube complexes. Geom. Funct. Anal. 17(2008) 1551–1620. http://dx.doi.Org/10.1007/s00039-007-0629-4 Google Scholar
[13] Ratcliffe, J. G., On normal subgroups of an amalgamated product of groups with applications to knot theory. Bol. Soc. Mat. Mex. 20(2014), no. 2, 287–296. http://dx.doi.Org/1 0.1007/s40590-014-0036-4 Google Scholar
[14] Scott, G. P. and Wall, C. T. C., Topological methods in group theory. In: Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., 36, Cambridge University Press, Cambridge-New York, 1979, pp. 137–203.Google Scholar
[15] Serre, J.-P., Trees. Springer-Verlag, Berlin-New York, 1980.Google Scholar
[16] Wise, D. T., A non-Hopfian automatic group. J. Algebra 180(1996), no. 3, 845–847. http://dx.doi.Org/10.1006/jabr.1996.0096 Google Scholar
[17] Wise, D. T., From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry. CBMS Regional Conference Series in Mathematics, 117, American Mathematical Society, Providence, RI, 2012. http://dx.doi.Org/10.1090/cbms/117 Google Scholar
[18] Wise, D. T., Cubular tubular groups. Trans. Amer. Math. Soc. 366(2014), no. 10, 5503–5521. http://dx.doi.Org/1 0.1090/S0002-9947-2014-06065-0 Google Scholar
[19] Woodhouse, D. J., Classifying finite dimensional cubulations of tubular groups. Michigan Math. J. 65(2016), 511–532. http://dx.doi.Org/10.1307/mmjV1472066145 Google Scholar
[20] Woodhouse, D. J., Classifying virtually special tubular groups. arxiv:1607.06334Google Scholar