Geometry of Defining Relations in Groups

It is clear that the relationship between algebra and geometry lacks lateral symmetry. The dialectics of their interaction is better reflected by the formula 'geometry - algebra - geometry' than by its mirror-image. Sufficiently convinc- ing historical examples are Descartes' coordinatization, Klein's Erlanger- programme, and the development of algebraic topology from Poincare's notion of fundamental group. It is true that there are some isolated examples which could be used to suggest the converse, but the typical pattern is the one-way traffic of algebraic concepts being used, and frequently arising, in the study of objects of a geometric or topological nature.
A special feature of this book is the systematic implementation of the non- standard formula 'algebra - geometry - algebra' resulting in the use of elemen- tary topological and geometric techniques for solving a number of problems that arise naturally in algebra. In this way we have obtained solutions to quite a few old problems in group theory. It should be stressed that we are speaking, as a rule, about general algebraic questions in areas that had already been developed without any appeal to geometric intuition. And conversely, we do not mention questions about groups generated by reflections, or other discrete groups of obvi- ously geometric origin, whose non-geometric study is virtually inconceivable.
A simple but important observation made by van Kampen [112] in 1933 remained unnoticed for more than three decades (unlike several other works of this prominent mathematician). The essence of van Kampen's lemma is the visual geometric interpretation of the process of deriving consequences of defining relations in a group. The idea of this method for depicting conse- quences is apparent from Figures la) and lb), where we exhibit the derivation of the consequences a 2 b 3 = b 3 a 2 from ab = ba, and of b 6 = 1 from b 2 = a and a 3 = 1: walking round each region of the map gives rise to one of the given relators (aba-1b- 1 , and b 2 a- 1 and a 3 , respectively) while the circuit of the
boundary of the whole map yields the consequences (a 2 b 3 a- 2 b- 3 and b 6 , respec- tively) provided that traversing an edge in the direction opposite to that indicated by the arrow gives the inverse letter.