Skip to main content Accessibility help

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

  • A. H. DOOLEY (a1) and V. YA. GOLODETS (a1)


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 ss′.



Hide All
[1]Adams, S.. Trees and amenable equivalence relations. Ergod. Th. & Dynam. Sys. 10 (1990), 114.
[2]Adams, S. and Spatzier, R.. Kazhdan groups, cocycles and trees. Amer. J. Math. 112 (1990), 271287.
[3]Brown, K. S.. Cohomology of Groups. Springer, New York, 1982.
[4]Cohen, D. E.. Groups with free subgroups of finite index. Conference on Group Theory (Lecture Notes in Mathematics, 319). Springer, Berlin, 1973, pp. 2644.
[5]Cohen, D. E.. Groups of Cohomological Dimension One (Lecture Notes in Mathematics, 245). Springer, Berlin, 1972.
[6]Cohen, D. E.. Combinatorial Group Theory: A Topological Approach. Cambridge University Press, Cambridge, 1989.
[7]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam Sys. 1 (1981), 431450.
[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.
[9]Dooley, A. H. and Golodets, V. Ya.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196 (2002), 118.
[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), 87124.
[11]Dyer, J. L. and Scott, G. P.. Periodic automorphisms of free groups. Commun. Algebra 3 (1975), 195201.
[12]Feldman, J. and Moore, C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234 (1977), 289324; II. 234 (1977), 325–359.
[13]Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math. 739 (2000), 4198.
[14]Gaboriau, D.. Invariants 2 de relations d’équivalence et de groupes. Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 93150.
[15]Gaboriau, D.. On orbit equivalence of measure preserving actions. Rigidity in Dynamics and Geometry (Cambridge, 2000). Springer, Berlin, 2002, pp. 167186.
[16]Gaboriau, D.. Examples of groups that are measure equivalent to the free group. Ergod. Th. & Dynam. Sys. 25 (2005), 18091827.
[17]Higman, G., Neumann, B. and Neumann, H.. Embedding theorems for groups. J. London Math. Soc. 14 (1949), 247257.
[18]Hjorth, G.. A lemma for cost attained. Ann. Pure Appl. Logic 143 (2006), 87102.
[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).
[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), 85153.
[21]Jackson, S., Kechris, A. S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Logic 2 (2002), 180.
[22]Karrass, A., Pietrowski, A. and Solitar, D.. Finitely and infinite cyclic extensions of free groups. J. Aust. Math. Soc. 16 (1973), 458466.
[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), 227255.
[24]Karrass, A. and Solitar, D.. Subgroups of HNN groups and groups with one defining relation. Canad. J. Math 23 (1971), 627643.
[25]Kechris, A. S. and Miller, B. D.. Topics in Orbit Equivalence Theory (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.
[26]Kirillov, A. A.. Elements of the Theory of Representations. Nauka, Moscow, 1972, 1978; English transl. Springer, Berlin, 1976.
[27]Lang, S.. Algebra. Addison-Wesley, Reading, MA, 1965.
[28]Levitt, G.. On the cost of generating an equivalence relation. Ergod. Th. & Dynam. Sys. 15 (1995), 11731181.
[29]Lyndon, R. and Schupp, R.. Combinatorial Group Theory, Band 89. Springer, Berlin, 1977.
[30]McCool, J.. A characterization of periodic automorphisms of a free group. Trans. Amer. Math. Soc. 260 (1980), 33093318.
[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. 494498.
[32]Ornstein, D. and Weiss, B.. Ergodic theory of amenable group actions, I. The Rohlin lemma. Bull. Amer. Math. Soc. 2 (1980), 161164.
[33]Pemantle, R. and Peres, Y.. Nonamenable products are not treeable. Israel J. Math. 118 (2004), 147155.
[34]Roman’kov, V. A.. Automorphisms of groups. Acta Appl. Math. 29 (1992), 241280.
[35]Schrier, O.. Die Untergruppen der freien Gruppen. Abh. Math. Sem. Univ. Hamburg 5 (1927), 161183.
[36]Scott, G. P.. An embedding theorem for groups with a free subgroup of finite index. Bull. London Math. Soc. 6 (1974), 304306.
[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. 137203.
[38]Serre, J.-P.. Sur la dimension cohomologique de groupes profinis. Topology 3 (1965), 413420.
[39]Serre, J.-P.. Trees. Springer, Berlin, 1980.
[40]Shalom, Y.. Measurable Group Theory, 4. ECM, Stockholm, 2004, pp. 391423.
[41]Stallings, J. R.. On torsion-free groups with infinitely many generators. Ann. Math. 88 (1968), 312334.
[42]Stallings, J. R.. Groups of cohomological dimension one. Amer. Math. Soc. (1970), 124128.
[43]Swan, R. G.. Groups of cohomological dimension one. J. Algebra 12 (1969), 585610.

Related content

Powered by UNSILO

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

  • A. H. DOOLEY (a1) and V. YA. GOLODETS (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.