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Large groups, property (τ) and the homology growth of subgroups

Published online by Cambridge University Press:  01 May 2009

MARC LACKENBY
MARC LACKENBY*
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB. e-mail: [email protected]
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Extract

We investigate the homology of finite index subgroups Gi of a given finitely presented group G. We fix a prime p, denote the field of order p by p and define dp(Gi) to be the dimension of H1(Gi; p). We will be interested in the situation where dp(Gi) grows fast as a function of the index [G : Gi]. Specifically, we say that a collection of finite index subgroups {Gi} has linear growth of mod p homology if infidp(Gi)/[G : Gi] is positive. This is a natural and interesting condition that arises in several different contexts. For example, the main theorem of [9] states that when G is a lattice in PSL(2, ℂ) with non-trivial torsion (equivalently, G is the fundamental group of a finite-volume hyperbolic 3-orbifold with non-empty singular locus), then G has such a sequence of subgroups. Another major class of groups G having such a collection of subgroups are those that are large. By definition, this means that G has a finite index subgroup that admits a surjective homomorphism onto a free non-abelian group. Large groups have many nice properties, for example super-exponential subgroup growth and infinite virtual first Betti number. One might wonder whether largeness is equivalent to the existence of some nested sequence of finite index subgroups {Gi} with linear growth of mod p homology for some prime p. If so, this would establish that lattices in PSL(2, ℂ) with non-trivial torsion are large, which would be a major breakthrough in low-dimensional topology.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 625 - 648
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Baumslag, G.Wreath products and finitely presented groups. Math. Z. 75 (1960/1961), 22–28.CrossRefGoogle Scholar
[2]Bourgain, J. and Gamburd, A.Uniform expansion bounds for Cayley graphs of SL(2, p). Ann. of Math. 167 (2008), no. 2, 625–642.CrossRefGoogle Scholar
[3]Bourgain, J., Gamburd, A. and Sarnak, P.Sieving and expanders. C. R. Math. Acad. Sci. Paris 343 (2006), no. 3, 155–159.CrossRefGoogle Scholar
[4]Clozel, L.Démonstration de la conjecture τ. Invent. Math. 151 (2003), no. 2, 297–328.CrossRefGoogle Scholar
[5]Ershov, M.Golod–Shafarevich groups with Property (T) and Kac–Moody groups. Duke. Math. J. 145 (2008), 309–339.CrossRefGoogle Scholar
[6]Justesen, J.A class of constructive asymptotically good algebraic codes. IEEE Trans. Inf. Theory IT-18 (1972), 652–656.CrossRefGoogle Scholar
[7]Lackenby, M.Heegaard splittings, the virtually Haken conjecture and property τ. Invent. Math. 164 (2006), 317–359.CrossRefGoogle Scholar
[8]Lackenby, M.Expanders, rank and graphs of groups. Israel J. Math. 146 (2005), 357–370.CrossRefGoogle Scholar
[9]Lackenby, M.Covering spaces of 3-orbifolds. Duke Math J. 136 (2007), 181–203.CrossRefGoogle Scholar
[10]Lubotzky, A.Discrete groups, expanding graphs and invariant measures. Prog. in Math. 125 (1994).Google Scholar
[11]Lubotzky, A.Free quotients and the first Betti number of some hyperbolic manifolds. Transform. Groups 1 (1996), no. 1-2, 71–82.CrossRefGoogle Scholar
[12]Lubotzky, A.Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem. Ann. of Math. 144 (1996), 441–452.CrossRefGoogle Scholar
[13]Lubotzky, A. and Pak, I.The product replacement algorithm and Kazhdan's property (T). J. Amer. Math. Soc. 14 (2001), 347–363.CrossRefGoogle Scholar
[14]Lubotzky, A. and Segal, D.Subgroup growth. Progr. in Math. 212 (2003).Google Scholar
[15]Lubotzky, A. and Weiss, B.Groups and expanders. Expanding graphs (Princeton, 1992), 95–109, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 10.CrossRefGoogle Scholar
[16]Lubotzky, A. and Zimmer, R.Variants of Kazhdan's property for subgroups of semisimple groups. Israel J. Math. 66 (1989), 289–299.CrossRefGoogle Scholar
[17]Spielman, D.Constructing error-correcting codes from expander graphs. Emerging applications of number theory (Minneapolis, 1996), 591–600, IMA Vol. Math. Appl. 109.CrossRefGoogle Scholar