NON-AMENABLE FINITELY PRESENTED TORSION-BY-CYCLIC GROUPS by ALEXANDER YU. OL’SHANSKII and MARK V. S A P I R ABSTRACT We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a fi- nitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent n 1 by a cyclic group, so it satisfies the identity [x, y]n = 1. CONTENTS 1 Introduction..................................................................................... 44 1.1 Shorthistoryoftheproblem.................................................................. 44 1.2 CongruenceextensionembeddingsofthefreeBurnsidegroups.................................... 46 1.3 TheschemeoftheproofofTheorem1.3andtheplanofthepaper................................ 50 2 Mapsanddiagrams.............................................................................. 53 2.1 Graphsandmaps........................................................................... 53 2.2 Bands..................................................................................... 54 2.3 Bondsandcontiguitysubmaps................................................................ 56 2.4 ConditionA................................................................................ 59 3 Burnsidequotients ............................................................................... 64 3.1 Axioms.................................................................................... 64 3.2 Corollariesof(Z1),(Z2),(Z3) ................................................................. 67 3.3 Theconstructionofagradedpresentation...................................................... 74 4 SubgroupsofthefreeBurnsidegroups.............................................................. 93 4.1 Setsofwordswithsmallcancellation........................................................... 93 4.2 Subgroups of B(m, n) satisfyingthecongruenceextensionproperty................................. 95 4.3 Burnsidegroupsandfreeproducts.............................................................104 5 Thepresentationofthegroup .....................................................................107 5.1 S-machinesandtheirinterpretation............................................................107 5.2 The S-machine M ..........................................................................111 5.3 Thepresentation............................................................................114 5.4 The subgroup A of H is a torsion group of exponent n ........................................115 6PropertiesoftheS-machine M .....................................................................116 6.1 The inverse semigroup P(M) .................................................................116 6.2 Stabilizers in P(M) ofsubwordsofthehub.....................................................123 6.3 Stabilizersofarbitraryadmissiblewords........................................................126 7 Bands,annuli,andtrapezia........................................................................142 7.1 Definitionsandbasicfacts....................................................................142 7.2 Forbiddenannuli............................................................................143 7.3 Trapezia...................................................................................144 8ThegroupHra ...................................................................................147 9ThegroupHkra ..................................................................................151 9.1 Getting rid of Burnside ra-cells................................................................151 9.2 Conditions(Z1),(Z2),(Z3)....................................................................159 10ProofofTheorem1.3.............................................................................165 References.........................................................................................166 Subjectindex.......................................................................................167 Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 99-01-00894 and by the INTAS grant, the research of the second author was supported in part by the NSF grant DMS 9978802. 44 ALEXANDER YU. OL’SHANSKII, MARK V. SAPIR 1. Introduction 1.1. Short history of the problem Hausdorff [16] proved in 1914 that one can subdivide the 2-sphere minus a countable set of points into 3 parts A, B, C, such that each of these three parts can be obtained from each of the other two parts by a rotation, and the union of two of these parts can be obtained by rotating the third part. This implied that one cannot define a finitely additive measure on the 2-sphere which is invariant under the group SO(3). In 1924 Banach and Tarski [4] generalized Hausdorff ’s result by proving, in particular, that in the Euclidean 3-space, every two bounded sets A, B with non-empty = n = n interiors can be decomposed A i=1 Ai, B i=1 Bi such that Ai can be rotated to Bi, i = 1,...,n (the so called Banach-Tarski paradox). Von Neumann [25] was first who noticed that the cause of the Banach-Tarski paradox is not the geometry of Eu- clidean 3-space but an algebraic property of the group SO(3). He introduced the con- cept of an amenable group (he called such groups “measurable”) as a group G which has a left invariant finitely additive measure1 µ, µ(G) = 1, noticed that if a group is amenable then any set it acts upon freely also has an invariant measure and proved that a group is not amenable provided it contains a free non-abelian subgroup. He also showed that groups like PSL(2, Z), SL(2, Z) contain free non-abelian subgroups. So analogs of Banach-Tarski paradox can be found in the Euclidean 2-space and even on the real line. Von Neumann showed that the class of amenable groups contains abelian groups, finite groups and is closed under taking subgroups, extensions, and in- finite unions of increasing sequences of groups. Day and Specht showed that this class is closed under homomorphic images. The class of groups without free non-abelian subgroups is also closed under these operations and contains abelian and finite groups. The problem of existence of non-amenable groups without non-abelian free sub- groups probably goes back to von Neumann and became known as the “von Neu- mann problem” in the fifties. As far as we know, the first paper where this problem was formulated was the paper by Day [10]. It is also mentioned in the monograph by Greenleaf [13] based on his lectures given in Berkeley in 1967. Tits [38] proved that every non-amenable matrix group over a field of characteristic 0 contains a non- abelian free subgroup. In particular every semisimple Lie group over a field of char- acteristic 0 contains such a subgroup (see also Vershik [39]). First counterexamples to the von Neumann problem were constructed by Ol’shanskii [29]. He proved that the groups with all proper subgroups cyclic con- structed by him, both torsion-free [27] and torsion [28] (the so called “Tarski mon- sters”), are not amenable. Later Adian [1] showed that the free Burnside group of odd 1 Later amenable groups were characterized in many different ways. There exist a geometric characterization by Følner [11], a characterization by Kesten [19] in terms of random walks, and a combinatorial characterization by Grigorchuk [14] among others. NON-AMENABLE FINITELY PRESENTED TORSION-BY-CYCLIC GROUPS 45 exponent n ≥ 665 with at least two generators (that is the group B(m, n) given by m generators and all relations of the form vn = 1 where v is a word in generators) is not amenable. It is interesting that the possibility of using torsion groups and, in particu- lar, free Burnside groups as potential counterexamples to von Neumann’s problem was mentioned by Day [10], conjectured by Kesten [20], and by B.H. Neumann’s in his review of Specht’s paper [37] in Zentralblatt. Both Ol’shanskii’s and Adian’s examples are not finitely presented: in the mod- ern terminology these groups are inductive limits of word hyperbolic groups, but they are not hyperbolic themselves. Since many mathematicians (especially topologists) are mostly interested in groups acting “nicely” on manifolds, it is natural to ask if there ex- ists a finitely presented non-amenable group without non-abelian free subgroups. This question was explicitly formulated, for example, by Grigorchuk in [22] and by Co- hen in [9]. This question is one of a series of similar questions about finding finitely presented “monsters”, i.e. groups with unusual properties. Probably the most famous problem in that series is the problem about finding a finitely presented infinite tor- sion group. Other similar problems ask for finitely presented divisible group (group where every element has roots of every degree), finitely presented Tarski monster, etc. In each case a finitely generated example can be constructed as a limit of hyperbolic groups (see [30]), and there is no hope to construct finitely presented examples as such limits. One difficulty in constructing a finitely presented non-amenable group with- out free non-abelian subgroups is that there are “very few” known finitely presented groups without free non-abelian subgroups. Most non-trivial examples are solvable or “almost” solvable (see [21]), and so they are amenable. The only known examples of finitely presented groups without free non-abelian subgroups for which the prob- lem of amenability