Abstract
In 1911 M. Dehn posed the word and conjugacy problems for groups in general and provided algorithms which solved these problems for the fundamental groups of closed orientable two-dimensional manifolds. A crucial feature of these groups is that (with trivial exceptions) they are defined by a single relator r with the property that if s is a cyclic conjugate of r or r-1, with s ≠ r-1, there is very little cancellation in forming the product rs. Dehn’s algorithms have been extended to large classes of groups possessing presentations in which the defining relations have a similar small cancellation property. At first, investigations were concerned with the solution of the word problem for groups G presented as small cancellation quotients of a free group F. The theory was subsequently extended to the case where F is a free product, a free product with amalgamation, or an HNN extension. Moreover, strong results were obtained about algebraic properties; for example, one can classify torsion elements and commuting elements in small cancellation quotients.
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© 2001 Springer-Verlag Berlin Heidelberg
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Lyndon, R.C., Schupp, P.E. (2001). Small Cancellation Theory. In: Combinatorial Group Theory. Classics in Mathematics, vol 89. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61896-3_5
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DOI: https://doi.org/10.1007/978-3-642-61896-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41158-1
Online ISBN: 978-3-642-61896-3
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