Abstract: The main geometric tool used in our approach to consequences of relations in groups is a finite map on the plane with a length function on paths defined combinatorially with a 'gradation' subject to a number of additional requirements dictated by properties of defining relations. The notion of a continuous mapping and, in particular, a path can usually be defined with the help of the ordinary Euclidean metric. The intuitive idea of the two-dimensional continuum at the level of the Jordan curve theorem is essentially sufficient for understanding our proofs. It will nevertheless be useful to give formal definitions of a number of topological concepts for two main reasons. Firstly, the use of a metric in our context looks somewhat artificial, and secondly, we will sometimes need to consider maps which are not necessarily simply connected, and even maps on the torus or on spheres with one or more holes.
Publication Year: 1991
Publication Date: 1991-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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