New estimates of odd exponents of infinite Burnside groups

Abstract

This article consists of two parts. The first part presents a detailed history of the long-term joint work (1960–1968) of the author and P.S. Novikov on the proof of the infiniteness of the free Burnside groups B(m, n) for odd periods n ≥ 4381 and m > 1 generators (Sections 1 and 2). In Sections 3–10 we survey several significant results obtained by the author and his successors using the Novikov-Adian theory and its various modifications. In the second part (Sections 11–15) we outline a new modification of the Novikov-Adian theory. The new modification allows us to decrease to n ≥ 101 the lower bound on the odd periods n for which one can prove the infiniteness of the free periodic groups B(m, n). We plan to publish a full proof of this new result in the journal Russian Mathematical Surveys.