Non-amenable finitely presented torsion-by-cyclic groups.

*(English)*Zbl 1050.20019The central aim of this investigation is to construct a finitely presented non-amenable group without any free non-cyclic subgroups. This is a new, finitely presented counterexample to von Neumann’s well-known problem. In the fifties von Neumann asked whether all amenable groups are without free non-cyclic subgroups [see M. M. Day, Ill. J. Math. 1, 509–544 (1957; Zbl 0078.29402); F. P. Greenleaf, Invariant means on topological groups and their applications (1969; Zbl 0174.19001)]. His question was motivated by results demonstrating many similarities between amenable groups and groups without free non-cyclic subgroups. For example: both the classes of amenable groups and of the groups without free non-cyclic subgroups contain all Abelian groups, finite groups, and they both are closed under the operations of taking subgroups, homomorphic images, extensions and infinite unions of increasing sequences of groups. Moreover, J. von Neumann proved that a group is not amenable if it contains a free non-cyclic subgroup [Fundam. 13, 73–116 (1929; JFM 55.0151.01)]. J. Tits proved that every non-amenable matrix group over a field of characteristic \(0\) contains a free non-cyclic subgroup [J. Algebra 20, 250–270 (1972; Zbl 0236.20032)].

The first counterexamples giving negative answers to the von Neumann problem were built by A. Yu. Ol’shanskii [Usp. Mat. Nauk 35, No. 4(214), 199–200 (1980; Zbl 0452.20032)], who proved that the “Tarski monsters” constructed by A. Yu. Ol’shanskii [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 309–321 (1980; Zbl 0475.20025), ibid. 43, 1328–1393 (1979; Zbl 0431.20027)] are non-amenable (they contain no free non-cyclic subgroups because all proper subgroups in these groups are cyclic). S. I. Adyan showed that the free Burnside group \(B(m,n)\) with \(m\geq 2\) generators and with odd exponent \(n\geq 665\) is non-amenable (it contains no free subgroups because it is of finite exponent) [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 6, 1139–1149 (1982; Zbl 0512.60012)]. The counterexamples of Ol’shanskij and of Adyan both are not finitely presented. Thus it is natural to ask whether there exists a finitely presented non-amenable group without free non-cyclic subgroups [see, for example, V. D. Mazurov and E. I. Khukhro, The Kourovka notebook. Unsolved problems in group theory. 15th ed. (2002; Zbl 0999.20002); J. M. Cohen, J. Funct. Anal. 48, 301–309 (1982; Zbl 0499.20023)].

The main result of the current paper is: Theorem 1.1. For every sufficiently large odd \(n\) there exists a finitely presented group \(G\) which satisfies the following conditions: (1) \(G\) is an ascending HNN-extension of a finitely generated infinite group of exponent \(n\). (2) \(G\) is an extension of a non-locally finite group of exponent \(n\) by an infinite cyclic group. (3) \(G\) contains a subgroup isomorphic to a free two-generator Burnside group \(B(2,n)\). (4) \(G\) is a non-amenable finitely presented group without free non-cyclic subgroups.

Notice that the fact that \(G\) is a non-amenable group follows from point (3) above, from the theorem of Adyan mentioned earlier, and from the fact that amenable groups are closed under taking subgroups. The fact that \(G\) is without free non-cyclic subgroups evidently follows from point (2) above. – Theorem 1.1. follows from the following two auxiliary theorems, which are of independent interest, and for the presentation of which we need the notion of CEP (CEP stands for “Congruence Extension Property”). A subgroup \(H\leq G\) is a CEP-subgroup in the group \(G\) if any homomorphism \(H\to H_1\) of \(H\) onto a group \(H_1\) can be continued to a homomorphism \(G\to G_1\) onto an overgroup \(G_1\) of \(H_1\). This is denoted by \(H\leq_{\text{CEP}}G\).

Theorem 1.2. For every sufficiently large odd \(n\) there exists a natural number \(s=s(n)\) such that the free Burnside group \(B(\infty,n)\) of infinite countable rank is CEP-embedded into a free Burnside group \(B(s,n)\) of rank \(s\).

For any group \(H\) denote \(H^n=\langle h^n\mid h\in H\rangle\).

Further: Theorem 1.3. Let \(n\) be a sufficiently large odd number. Then for every positive integer \(s\) there is a finitely presented group \(H\) containing a subgroup \(B(s,n)\) such that: (1) \(H^n\cap B(s,n)=1\), so that there is a canonical embedding of \(B(s,n)\) into \(H/H^n\). (2) The embedding of \(B(s,n)\) into \(H/H^n\) from (1) satisfies CEP.

Since \(B(s,n)\) is non-amenable, there remains to transform its embedding into the finitely presented group \(H\) to such an embedding of \(B(s,n)\) into a group \(G\) which still is finitely presented, and which is an extension of a group of finite exponent by a cyclic group (and, thus, cannot contain a free non-cyclic subgroup). Assume \(H=\langle c_1,\dots,c_m\mid R \rangle\). Then \(H(n)=H/H^n=\langle c_1,\dots,c_m\mid R\cup V\rangle\), where \(V\) is the set of all \(n\)-th powers \(v^n(c_1,\dots,c_m)\) of words \(v\) on variables \(c_1,\dots,c_m\). Assume \(B_b(m,n)\) is the isomorphic copy of the group \(B(m,n)\) embedded into \(H\) by Theorem 1.2. Then for its generators \(b_i\) we have: \(b_i=w_i(c_1,\dots,c_m)\), \(i=1,\dots,m\). Denote by \(R'\) the set of words \(r', r'',\dots\) obtained from the words \(r(c_1,\dots,c_m)\in R\) by substitutions as follows: \[ r'(c_1,\dots,c_m)=r(w_1(c_1,\dots,c_m),\dots,w_m(c_1,\ldots ,c_m)); \] \[ r''(c_1,\dots,c_m)=r'(w_1(c_1,\dots,c_m),\ldots,w_m(c_1,\dots ,c_m)); \] etc. One can prove that the group \(\overline H=H(n)/(R')^{H(n)}=\langle c_1,\dots,c_m\mid R\cup V\cup R'\rangle\) is isomorphic to its image under the mapping \(c_i\to b_i\), \(i=1,\dots,m\). This isomorphism defines the ascending HNN-extension \(G=\langle\overline H ,t\mid c_i^t=b_i\), \(i=1,\dots,m\rangle\) which turns out to be finitely presented because all its defining relations follow from the finite set \(R\) and from the finite set of relations \(c_i^t=w_i\), \(i=1,\dots,m\).

The main technical argument of the paper is the proof of Theorem 1.3. Its group \(H(n)\) is the Burnside factor \(H/H^n\) of the group \(H\). The construction of the latter is similar to that of the authors [Contemp. Math. 264, 23–47 (2000; Zbl 0987.20015), Int. J. Algebra Comput. 11, No. 2, 137–170 (2001; Zbl 1025.20030)]; M. V. Sapir, J.-C. Birget and E. Rips [Ann. Math. (2) 156, No. 2, 345–466 (2002; Zbl 1026.20021)]. In particular, an S-machine is being used. It is an analog of the well known Turing machine, but it works not with elements of a free semigroup but with elements of a group.

The paper is very well written and, thanks to its detailed opening sections, is a good source to study a few modern directions in combinatorial group theory.

The first counterexamples giving negative answers to the von Neumann problem were built by A. Yu. Ol’shanskii [Usp. Mat. Nauk 35, No. 4(214), 199–200 (1980; Zbl 0452.20032)], who proved that the “Tarski monsters” constructed by A. Yu. Ol’shanskii [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 309–321 (1980; Zbl 0475.20025), ibid. 43, 1328–1393 (1979; Zbl 0431.20027)] are non-amenable (they contain no free non-cyclic subgroups because all proper subgroups in these groups are cyclic). S. I. Adyan showed that the free Burnside group \(B(m,n)\) with \(m\geq 2\) generators and with odd exponent \(n\geq 665\) is non-amenable (it contains no free subgroups because it is of finite exponent) [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 6, 1139–1149 (1982; Zbl 0512.60012)]. The counterexamples of Ol’shanskij and of Adyan both are not finitely presented. Thus it is natural to ask whether there exists a finitely presented non-amenable group without free non-cyclic subgroups [see, for example, V. D. Mazurov and E. I. Khukhro, The Kourovka notebook. Unsolved problems in group theory. 15th ed. (2002; Zbl 0999.20002); J. M. Cohen, J. Funct. Anal. 48, 301–309 (1982; Zbl 0499.20023)].

The main result of the current paper is: Theorem 1.1. For every sufficiently large odd \(n\) there exists a finitely presented group \(G\) which satisfies the following conditions: (1) \(G\) is an ascending HNN-extension of a finitely generated infinite group of exponent \(n\). (2) \(G\) is an extension of a non-locally finite group of exponent \(n\) by an infinite cyclic group. (3) \(G\) contains a subgroup isomorphic to a free two-generator Burnside group \(B(2,n)\). (4) \(G\) is a non-amenable finitely presented group without free non-cyclic subgroups.

Notice that the fact that \(G\) is a non-amenable group follows from point (3) above, from the theorem of Adyan mentioned earlier, and from the fact that amenable groups are closed under taking subgroups. The fact that \(G\) is without free non-cyclic subgroups evidently follows from point (2) above. – Theorem 1.1. follows from the following two auxiliary theorems, which are of independent interest, and for the presentation of which we need the notion of CEP (CEP stands for “Congruence Extension Property”). A subgroup \(H\leq G\) is a CEP-subgroup in the group \(G\) if any homomorphism \(H\to H_1\) of \(H\) onto a group \(H_1\) can be continued to a homomorphism \(G\to G_1\) onto an overgroup \(G_1\) of \(H_1\). This is denoted by \(H\leq_{\text{CEP}}G\).

Theorem 1.2. For every sufficiently large odd \(n\) there exists a natural number \(s=s(n)\) such that the free Burnside group \(B(\infty,n)\) of infinite countable rank is CEP-embedded into a free Burnside group \(B(s,n)\) of rank \(s\).

For any group \(H\) denote \(H^n=\langle h^n\mid h\in H\rangle\).

Further: Theorem 1.3. Let \(n\) be a sufficiently large odd number. Then for every positive integer \(s\) there is a finitely presented group \(H\) containing a subgroup \(B(s,n)\) such that: (1) \(H^n\cap B(s,n)=1\), so that there is a canonical embedding of \(B(s,n)\) into \(H/H^n\). (2) The embedding of \(B(s,n)\) into \(H/H^n\) from (1) satisfies CEP.

Since \(B(s,n)\) is non-amenable, there remains to transform its embedding into the finitely presented group \(H\) to such an embedding of \(B(s,n)\) into a group \(G\) which still is finitely presented, and which is an extension of a group of finite exponent by a cyclic group (and, thus, cannot contain a free non-cyclic subgroup). Assume \(H=\langle c_1,\dots,c_m\mid R \rangle\). Then \(H(n)=H/H^n=\langle c_1,\dots,c_m\mid R\cup V\rangle\), where \(V\) is the set of all \(n\)-th powers \(v^n(c_1,\dots,c_m)\) of words \(v\) on variables \(c_1,\dots,c_m\). Assume \(B_b(m,n)\) is the isomorphic copy of the group \(B(m,n)\) embedded into \(H\) by Theorem 1.2. Then for its generators \(b_i\) we have: \(b_i=w_i(c_1,\dots,c_m)\), \(i=1,\dots,m\). Denote by \(R'\) the set of words \(r', r'',\dots\) obtained from the words \(r(c_1,\dots,c_m)\in R\) by substitutions as follows: \[ r'(c_1,\dots,c_m)=r(w_1(c_1,\dots,c_m),\dots,w_m(c_1,\ldots ,c_m)); \] \[ r''(c_1,\dots,c_m)=r'(w_1(c_1,\dots,c_m),\ldots,w_m(c_1,\dots ,c_m)); \] etc. One can prove that the group \(\overline H=H(n)/(R')^{H(n)}=\langle c_1,\dots,c_m\mid R\cup V\cup R'\rangle\) is isomorphic to its image under the mapping \(c_i\to b_i\), \(i=1,\dots,m\). This isomorphism defines the ascending HNN-extension \(G=\langle\overline H ,t\mid c_i^t=b_i\), \(i=1,\dots,m\rangle\) which turns out to be finitely presented because all its defining relations follow from the finite set \(R\) and from the finite set of relations \(c_i^t=w_i\), \(i=1,\dots,m\).

The main technical argument of the paper is the proof of Theorem 1.3. Its group \(H(n)\) is the Burnside factor \(H/H^n\) of the group \(H\). The construction of the latter is similar to that of the authors [Contemp. Math. 264, 23–47 (2000; Zbl 0987.20015), Int. J. Algebra Comput. 11, No. 2, 137–170 (2001; Zbl 1025.20030)]; M. V. Sapir, J.-C. Birget and E. Rips [Ann. Math. (2) 156, No. 2, 345–466 (2002; Zbl 1026.20021)]. In particular, an S-machine is being used. It is an analog of the well known Turing machine, but it works not with elements of a free semigroup but with elements of a group.

The paper is very well written and, thanks to its detailed opening sections, is a good source to study a few modern directions in combinatorial group theory.

Reviewer: Vahagn H. Mikaelian (Yerevan)

##### MSC:

20F05 | Generators, relations, and presentations of groups |

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

20F50 | Periodic groups; locally finite groups |

20E07 | Subgroup theorems; subgroup growth |

43A07 | Means on groups, semigroups, etc.; amenable groups |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20A15 | Applications of logic to group theory |