Combinatorial Estimations on Burnside Type Problems

The Burnside problem, formulated by W. Burnside in 1902, is one of the most well-known and important open questions in the field of Group Theory. Despite significant progress made in the past century towards solving this problem, its complete solution remains unknown. In this paper, we investigate one of the approaches to solving the Burnside problem based on the application of an iterative theory of small cancellations and canonical forms developed by E. Rips in recent years. We present a novel self-contained exposition of this theory and utilize it to obtain new estimates on the infiniteness of initial approximations of Burnside groups where only a finite number of periodic relations is used for relatively small odd exponents (n>120).