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Detecting ends of residually finite groups in profinite completions

Published online by Cambridge University Press:  22 August 2013

OWEN COTTON–BARRATT
OWEN COTTON–BARRATT*
Affiliation:
Mathematical Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ. e-mail: [email protected]
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Abstract

Let $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for $\mathcal{C}$ the class of finite p-groups.

We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 379 - 389
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Aka, M. Profinite completions and Kazhdan's property (T). Arxiv preprint arXiv:1005.4566 (2010).Google Scholar
[2]Arzhantseva, G., Bridson, M. R., Januszkiewicz, T., Leary, I. J., Minasyan, A., and Brodzki, J.Infinite groups with fixed point properties. Geom. Topol. 13 (3) (2009), 1229–1263.CrossRefGoogle Scholar
[3]Baumslag, G.Groups with the same lower central sequence as a relatively free group. II. Properties. Trans. Amer. Math. Soc. 142 (1969), 507–538.CrossRefGoogle Scholar
[4]Belegradek, I. and Osin, D.Rips construction and Kazhdan property (T). Groups Geom. Dyn. 2 (1) (2008), 1–12.CrossRefGoogle Scholar
[5]Bridson, M. R.Decision problems and profinite completions of groups. J. Algebra 326 (2011), 59–73.CrossRefGoogle Scholar
[6]Bridson, M. R. and Grunewald, F. J.Grothendieck's problems concerning profinite completions and representations of groups. Ann. of Math. (2), 160 (1) (2004), 359–373.CrossRefGoogle Scholar
[7]Bridson, M. R. and Haefliger, A.Metric spaces of non-positive curvature, 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. (Springer-Verlag, Berlin, 1999).Google Scholar
[8]Bridson, M. R. and Wilton, H. The triviality problem for profinite completions. In preparation.Google Scholar
[9]Cotton-Barratt, O. and Wilton, H.Conjugacy separability of 1-acylindrical graphs of free groups. Math. Z. (to appear). 10.1007/s00209-012-0978-z.Google Scholar
[10]Gildenhuys, D. and Ribes, L.Profinite groups and Boolean graphs. J. Pure Appl. Algebra 12 (1) (1978), 21–47.CrossRefGoogle Scholar
[11]Grunewald, F. and Zalesskii, P.Genus for groups. J. Algebra 326 (2011), 130–168.CrossRefGoogle Scholar
[12]Haglund, F. and Wise, D. T.Special cube complexes. Geom. Funct. Anal. 17 (5) (2008), 1551–1620.CrossRefGoogle Scholar
[13]Kassabov, M. Pro-finite properties and property tau. Unpublished talk, 2008.Google Scholar
[14]Lackenby, M.Detecting large groups. J. Algebra 324 (10) (2010), 2636–2657.CrossRefGoogle Scholar
[15]Martino, A. and Minasyan, A. Conjugacy in normal subgroups of hyperbolic groups. Preprint (2009).Google Scholar
[16]Menny, A. Profinite completions and Kazhdan's property (T). Arxiv preprint arXiv:1005.4566 (2010).Google Scholar
[17]Nikolov, N. and Segal, D.On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. of Math. (2) 165 (1) (2007), 171–238.CrossRefGoogle Scholar
[18]Ribes, L. and Zalesskii, P. A.Conjugacy separability of amalgamated free products of groups. J. Algebra 179 (3) (1996), 751–774.CrossRefGoogle Scholar
[19]Rips, E.Subgroups of small cancellation groups. Bull. London Math. Soc. 14 (1) (1982), 45–47.CrossRefGoogle Scholar
[20]Serre, J.Arbres, amalgames, SL2. Société Mathématique de France, Paris, 1977. Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.Google Scholar
[21]Stallings, J.On torsion-free groups with infinitely many ends. Ann. of Math. (2), 88 (1968), 312–334.CrossRefGoogle Scholar
[22]Wise, D. T.A residually finite version of Rips's construction. Bull. London Math. Soc. 35 (1) (2003), 23–29.CrossRefGoogle Scholar
[23]Zalesskiĭ, P. A.Geometric characterization of free constructions of profinite groups. Sibirsk. Mat. Zh. 30 (2) (1989), 73–84, 226.Google Scholar
[24]Zalesskiĭ, P. A. and Mel'nikov, O.V.Subgroups of profinite groups acting on trees. Mat. Sb. (N.S.) 135 (177)(4) (1988), 419–439, 559.Google Scholar
[25]Zalesskiĭ, P. A. and Mel'nikov, O.V.Fundamental groups of graphs of profinite groups. Algebra i Analiz 1 (4) (1989), 117–135.Google Scholar