New estimates of odd exponents of infinite Burnside groups.

*(English. Russian original)*Zbl 1343.20040
Proc. Steklov Inst. Math. 289, 33-71 (2015); translation from Tr. Mat. Inst. Steklova 289, 41-82 (2015).

This paper is divided in two parts. The first one is a historical survey of the proof of the infiniteness of the free Burnside groups \(\mathbf B(m,n)\) on \(m>1\) generators and odd exponent \(n\geq 4381\). Of particular interest is the description of the genesis of the proof: “In the spring of 1958, Yu. I. Sorkin, a student of A. G. Kurosh, gave a talk on a result saying that in two-letter alphabet one can construct an infinite sequence containing no cube of any word. The first proof of this result was published by A. Thue in 1906.”

“After Sorkin’s talk, Novikov expressed an idea that one could try to use such sequence in order to prove infiniteness of the periodic groups \(\mathbf B(m,n)\) for some large values of the exponent \(n\). For that purposes it would be sufficient to find in these groups \(\mathbf B(m,n)\) a system of defining relations of the form \(A^n_i\equiv 1\) such that the left-hand sides of any pair of different relations intersect along a sufficiently small piece of each of them (say, less than \(1/6\) of the length). Then, according to the well-known lemma by Greendlinger and Tartakovskii [M. Greendlinger, Sov. Math., Dokl. 5, 110-112 (1964; Zbl 0134.25905); translation from Dokl. Akad. Nauk SSSR 154, 507-509 (1964)], all words in the Thue sequence of cube-free words would represent different elements of the group \(\mathbf B(2,n)\). However, the possibility of presenting the group \(\mathbf B(2m,n)\) by a system of defining relations satisfying such a condition for any pair of relations looks nonrealistic. So it was necessary to find another proof of the following desired statement: If a reduced word \(W\) is equal to \(1\) in \(\mathbf B(m,n)\), then it must contain a subword of the form \(A^3\).”

In the second part of the paper, the author outlines a modification of the Novikov-Adian theory in order to decrease to \(n\geq 101\) the lower bound on the odd exponents \(n\) for which one can prove the infiniteness of \(\mathbf B(m,n)\).

“After Sorkin’s talk, Novikov expressed an idea that one could try to use such sequence in order to prove infiniteness of the periodic groups \(\mathbf B(m,n)\) for some large values of the exponent \(n\). For that purposes it would be sufficient to find in these groups \(\mathbf B(m,n)\) a system of defining relations of the form \(A^n_i\equiv 1\) such that the left-hand sides of any pair of different relations intersect along a sufficiently small piece of each of them (say, less than \(1/6\) of the length). Then, according to the well-known lemma by Greendlinger and Tartakovskii [M. Greendlinger, Sov. Math., Dokl. 5, 110-112 (1964; Zbl 0134.25905); translation from Dokl. Akad. Nauk SSSR 154, 507-509 (1964)], all words in the Thue sequence of cube-free words would represent different elements of the group \(\mathbf B(2,n)\). However, the possibility of presenting the group \(\mathbf B(2m,n)\) by a system of defining relations satisfying such a condition for any pair of relations looks nonrealistic. So it was necessary to find another proof of the following desired statement: If a reduced word \(W\) is equal to \(1\) in \(\mathbf B(m,n)\), then it must contain a subword of the form \(A^3\).”

In the second part of the paper, the author outlines a modification of the Novikov-Adian theory in order to decrease to \(n\geq 101\) the lower bound on the odd exponents \(n\) for which one can prove the infiniteness of \(\mathbf B(m,n)\).

Reviewer: Enrico Jabara (Venezia)

##### MSC:

20F50 | Periodic groups; locally finite groups |

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

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

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

20-03 | History of group theory |

01A60 | History of mathematics in the 20th century |

##### Keywords:

periodic groups; free Burnside groups; Burnside problem; presentations of groups; word problem; groups of odd exponent
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\textit{S. I. Adian}, Proc. Steklov Inst. Math. 289, 33--71 (2015; Zbl 1343.20040); translation from Tr. Mat. Inst. Steklova 289, 41--82 (2015)

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