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Related Experiment Videos

ON A GENERALIZATION OF DEHN'S ALGORITHM.

Oliver Goodman1, Michael Shapiro

  • 1Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3052, Australia.

International Journal of Algebra and Computation
|December 17, 2009
PubMed
Summary
This summary is machine-generated.

This study generalizes Dehn's algorithm for solving word problems in group theory. The enhanced algorithm applies to new classes of groups, including nilpotent and hyperbolic groups, with specific exceptions noted.

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Area of Science:

  • Group Theory
  • Computational Group Theory
  • Algebraic Topology

Background:

  • Dehn's algorithm is a fundamental tool for solving the word problem in certain groups.
  • Existing algorithms have limitations in the types of groups they can address.

Purpose of the Study:

  • To generalize Dehn's algorithm for solving the word problem.
  • To extend the applicability of the algorithm to a broader range of groups.
  • To identify conditions under which such algorithms cannot exist.

Main Methods:

  • Viewing Dehn's algorithm as a rewriting system.
  • Generalizing the alphabet to include non-group elements.
  • Analyzing group properties like nilpotency and hyperbolicity.

Main Results:

  • The generalized algorithm solves the word problem for finitely generated nilpotent groups.
  • It also applies to many relatively hyperbolic groups, including geometrically finite and certain 3-manifold fundamental groups.
  • Groups with commuting infinite subgroups of exponential growth do not admit such algorithms (Cannon's algorithms).

Conclusions:

  • The generalized algorithm expands the scope of solvable word problems in group theory.
  • The identified exceptions (Cannon's algorithms) highlight fundamental limitations.
  • The class of groups solvable by this method exhibits useful closure properties.