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The geometric dimension of an equivalence relation and finite extensions of countable groups

Published online by Cambridge University Press:  02 March 2009

A. H. DOOLEY  and
V. YA. GOLODETS
A. H. DOOLEY
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au, v.golodets@unsw.edu.au)
V. YA. GOLODETS
Affiliation:
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au, v.golodets@unsw.edu.au)
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Abstract

We say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1789 - 1814
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Adams, S.. Trees and amenable equivalence relations. Ergod. Th. & Dynam. Sys. 10 (1990), 1–14.CrossRefGoogle Scholar
[2]Adams, S. and Spatzier, R.. Kazhdan groups, cocycles and trees. Amer. J. Math. 112 (1990), 271–287.CrossRefGoogle Scholar
[3]Brown, K. S.. Cohomology of Groups. Springer, New York, 1982.CrossRefGoogle Scholar
[4]Cohen, D. E.. Groups with free subgroups of finite index. Conference on Group Theory (Lecture Notes in Mathematics, 319). Springer, Berlin, 1973, pp. 26–44.CrossRefGoogle Scholar
[5]Cohen, D. E.. Groups of Cohomological Dimension One (Lecture Notes in Mathematics, 245). Springer, Berlin, 1972.CrossRefGoogle Scholar
[6]Cohen, D. E.. Combinatorial Group Theory: A Topological Approach. Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
[7]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam Sys. 1 (1981), 431–450.CrossRefGoogle Scholar
[8]Dooley, A. H. and Golodets, V. Ya.. The cost of an equivalence relation is determined by the cost of a finite index subrelation, submitted.Google Scholar
[9]Dooley, A. H. and Golodets, V. Ya.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196 (2002), 1–18.CrossRefGoogle Scholar
[10]Dooley, A. H., Ya. Golodets, V., Rudolph, D. J. and Sinel’shchikov, S. D.. Non-Bernoulli systems with completely positive entropy. Ergod. Th. & Dynam Sys. 28 (2008), 87–124.CrossRefGoogle Scholar
[11]Dyer, J. L. and Scott, G. P.. Periodic automorphisms of free groups. Commun. Algebra 3 (1975), 195–201.CrossRefGoogle Scholar
[12]Feldman, J. and Moore, C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234 (1977), 289–324; II. 234 (1977), 325–359.CrossRefGoogle Scholar
[13]Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math. 739 (2000), 41–98.CrossRefGoogle Scholar
[14]Gaboriau, D.. Invariants ℓ 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 93–150.CrossRefGoogle Scholar
[15]Gaboriau, D.. On orbit equivalence of measure preserving actions. Rigidity in Dynamics and Geometry (Cambridge, 2000). Springer, Berlin, 2002, pp. 167–186.CrossRefGoogle Scholar
[16]Gaboriau, D.. Examples of groups that are measure equivalent to the free group. Ergod. Th. & Dynam. Sys. 25 (2005), 1809–1827.CrossRefGoogle Scholar
[17]Higman, G., Neumann, B. and Neumann, H.. Embedding theorems for groups. J. London Math. Soc. 14 (1949), 247–257.CrossRefGoogle Scholar
[18]Hjorth, G.. A lemma for cost attained. Ann. Pure Appl. Logic 143 (2006), 87–102.CrossRefGoogle Scholar
[19]Hjorth, G. and Kechris, A. S.. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Amer. Math. Soc. 177(833) (2005).Google Scholar
[20]Ionna, A., Peterson, J. and Popa, S.. Amalgamated free products of w-rigid factors and calculation of their symmetry groups. Acta Math. 200 (2008), 85–153.CrossRefGoogle Scholar
[21]Jackson, S., Kechris, A. S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Logic 2 (2002), 1–80.CrossRefGoogle Scholar
[22]Karrass, A., Pietrowski, A. and Solitar, D.. Finitely and infinite cyclic extensions of free groups. J. Aust. Math. Soc. 16 (1973), 458–466.CrossRefGoogle Scholar
[23]Karrass, A. and Solitar, D.. The subgroups of a free product of two groups with an amalgamated subgroup. Trans. Amer. Math. Soc. 150 (1970), 227–255.CrossRefGoogle Scholar
[24]Karrass, A. and Solitar, D.. Subgroups of HNN groups and groups with one defining relation. Canad. J. Math 23 (1971), 627–643.CrossRefGoogle Scholar
[25]Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence Theory (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.CrossRefGoogle Scholar
[26]Kirillov, A. A.. Elements of the Theory of Representations. Nauka, Moscow, 1972, 1978; English transl. Springer, Berlin, 1976.Google Scholar
[27]Lang, S.. Algebra. Addison-Wesley, Reading, MA, 1965.Google Scholar
[28]Levitt, G.. On the cost of generating an equivalence relation. Ergod. Th. & Dynam. Sys. 15 (1995), 1173–1181.CrossRefGoogle Scholar
[29]Lyndon, R. and Schupp, R.. Combinatorial Group Theory, Band 89. Springer, Berlin, 1977.Google Scholar
[30]McCool, J.. A characterization of periodic automorphisms of a free group. Trans. Amer. Math. Soc. 260 (1980), 3309–3318.CrossRefGoogle Scholar
[31]Meskin, S.. Periodic automorphisms of the two-generator free group. Int. Conf. Theory of Groups (Lecture Notes in Mathematics, 372). Springer, Berlin, 1973, pp. 494–498.Google Scholar
[32]Ornstein, D. and Weiss, B.. Ergodic theory of amenable group actions, I. The Rohlin lemma. Bull. Amer. Math. Soc. 2 (1980), 161–164.CrossRefGoogle Scholar
[33]Pemantle, R. and Peres, Y.. Nonamenable products are not treeable. Israel J. Math. 118 (2004), 147–155.CrossRefGoogle Scholar
[34]Roman’kov, V. A.. Automorphisms of groups. Acta Appl. Math. 29 (1992), 241–280.CrossRefGoogle Scholar
[35]Schrier, O.. Die Untergruppen der freien Gruppen. Abh. Math. Sem. Univ. Hamburg 5 (1927), 161–183.CrossRefGoogle Scholar
[36]Scott, G. P.. An embedding theorem for groups with a free subgroup of finite index. Bull. London Math. Soc. 6 (1974), 304–306.CrossRefGoogle Scholar
[37]Scott, G. P. and Wall, C. T. C.. Topological Methods in Group Theory, Homological Group Theory (London Mathematical Society Lecture Notes, 36). Cambridge University Press, Cambridge, 1979, pp. 137–203.CrossRefGoogle Scholar
[38]Serre, J.-P.. Sur la dimension cohomologique de groupes profinis. Topology 3 (1965), 413–420.CrossRefGoogle Scholar
[39]Serre, J.-P.. Trees. Springer, Berlin, 1980.CrossRefGoogle Scholar
[40]Shalom, Y.. Measurable Group Theory, 4. ECM, Stockholm, 2004, pp. 391–423.Google Scholar
[41]Stallings, J. R.. On torsion-free groups with infinitely many generators. Ann. Math. 88 (1968), 312–334.CrossRefGoogle Scholar
[42]Stallings, J. R.. Groups of cohomological dimension one. Amer. Math. Soc. (1970), 124–128.Google Scholar
[43]Swan, R. G.. Groups of cohomological dimension one. J. Algebra 12 (1969), 585–610.CrossRefGoogle Scholar